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Flatland by Edwin A. Abbott
The Inhabitance of SPACE IN GENERAL
And H.C. IN PARTICULAR
This Work is Dedicated
By a Humble Native of Flatland
In the Hope that
Even as he was Initiated into the Mysteries
Of THREE DIMENSIONS
Having been previously conversant
With ONLY TWO
So the Citizens of that Celestial Region
May aspire yet higher and higher
To the Secrets of FOUR FIVE or EVEN SIX Dimensions
To the Enlargment of THE IMAGINATION
And the possible Development
Of that most and excellent Gift of MODESTY
Among the Superior Races
Of SOLID HUMANITY
SECTION 1 Of the Nature of Flatland
I call our world Flatland, not because we call it so,
but to make its nature clearer to you, my happy readers,
who are privileged to live in Space.
Imagine a vast sheet of paper on which straight Lines,
Triangles, Squares, Pentagons, Hexagons, and other figures,
instead of remaining fixed in their places, move freely about,
on or in the surface, but without the power of rising above
or sinking below it, very much like shadows--only hard
with luminous edges--and you will then have a pretty correct
notion of my country and countrymen. Alas, a few years ago,
I should have said "my universe:" but now my mind has been
opened to higher views of things.
In such a country, you will perceive at once that it is
impossible that there should be anything of what you call
a "solid" kind; but I dare say you will suppose that
we could at least distinguish by sight the Triangles, Squares,
and other figures, moving about as I have described them.
On the contrary, we could see nothing of the kind,
not at least so as to distinguish one figure from another.
Nothing was visible, nor could be visible, to us,
except Straight Lines; and the necessity of this
I will speedily demonstrate.
Place a penny on the middle of one of your tables in Space; and
leaning over it, look down upon it. It will appear a circle.
But now, drawing back to the edge of the table, gradually lower
your eye (thus bringing yourself more and more into the condition
of the inhabitants of Flatland), and you will find the penny
becoming more and more oval to your view, and at last when you
have placed your eye exactly on the edge of the table
(so that you are, as it were, actually a Flatlander)
the penny will then have ceased to appear oval at all,
and will have become, so far as you can see, a straight line.
The same thing would happen if you were to treat
in the same way a Triangle, or a Square, or any other figure
cut out from pasteboard. As soon as you look at it with your eye
on the edge of the table, you will find that it ceases to appear
to you as a figure, and that it becomes in appearance a straight line.
Take for example an equilateral Triangle--who represents with us
a Tradesman of the respectable class. Figure 1 represents
the Tradesman as you would see him while you were bending over
him from above; figures 2 and 3 represent the Tradesman,
as you would see him if your eye were close to the level,
or all but on the level of the table; and if your eye were
quite on the level of the table (and that is how we see him
in Flatland) you would see nothing but a straight line.
When I was in Spaceland I heard that your sailors
have very similar experiences while they traverse
your seas and discern some distant island or coast
lying on the horizon. The far-off land may have bays,
forelands, angles in and out to any number and extent;
yet at a distance you see none of these (unless indeed
your sun shines bright upon them revealing the projections
and retirements by means of light and shade), nothing but
a grey unbroken line upon the water.
Well, that is just what we see when one of our triangular
or other acquaintances comes towards us in Flatland.
As there is neither sun with us, nor any light of such
a kind as to make shadows, we have none of the helps
to the sight that you have in Spaceland.
If our friend comes closer to us we see
his line becomes larger; if he leaves us
it becomes smaller; but still he looks like
a straight line; be he a Triangle, Square,
Pentagon, Hexagon, Circle, what you will--
a straight Line he looks and nothing else.
You may perhaps ask how under these disadvantagous circumstances
we are able to distinguish our friends from one another:
but the answer to this very natural question will be more fitly
and easily given when I come to describe the inhabitants of Flatland.
For the present let me defer this subject, and say a word or two
about the climate and houses in our country.
SECTION 2 Of the Climate and Houses in Flatland
As with you, so also with us, there are four points
of the compass North, South, East, and West.
There being no sun nor other heavenly bodies, it is impossible
for us to determine the North in the usual way; but we have
a method of our own. By a Law of Nature with us,
there is a constant attraction to the South;
and, although in temperate climates this is very slight--
so that even a Woman in reasonable health can journey
several furlongs northward without much difficulty--
yet the hampering effort of the southward attraction
is quite sufficient to serve as a compass in most parts
of our earth. Moreover, the rain (which falls
at stated intervals) coming always from the North,
is an additional assistance; and in the towns
we have the guidance of the houses,
which of course have their side-walls
running for the most part North and South,
so that the roofs may keep off the rain from the North.
In the country, where there are no houses,
the trunks of the trees serve as some sort of guide.
Altogether, we have not so much difficulty as might
be expected in determining our bearings.
Yet in our more temperate regions, in which
the southward attraction is hardly felt,
walking sometimes in a perfectly desolate plain
where there have been no houses nor trees to guide me,
I have been occasionally compelled to remain stationary
for hours together, waiting till the rain came
before continuing my journey. On the weak and aged,
and especially on delicate Females, the force of attraction
tells much more heavily than on the robust of the Male Sex,
so that it is a point of breeding, if you meet a Lady on the street,
always to give her the North side of the way--by no means
an easy thing to do always at short notice when you are
in rude health and in a climate where it is difficult
to tell your North from your South.
Windows there are none in our houses: for the light
comes to us alike in our homes and out of them,
by day and by night, equally at all times and in all places,
whence we know not. It was in old days, with our learned men,
an interesting and oft-investigate question,
"What is the origin of light?" and the solution of it
has been repeatedly attempted, with no other result
than to crowd our lunatic asylums with the would-be solvers.
Hence, after fruitless attempts to suppress such investigations
indirectly by making them liable to a heavy tax, the Legislature,
in comparatively recent times, absolutely prohibited them.
I--alas, I alone in Flatland--know now only too well
the true solution of this mysterious problem;
but my knowledge cannot be made intelligible
to a single one of my countrymen; and I am mocked at
--I, the sole possessor of the truths of Space
and of the theory of the introduction of Light
from the world of three Dimensions--as if I were
the maddest of the mad! But a truce to these painful
digressions: let me return to our homes.
The most common form for the construction of a house
is five-sided or pentagonal, as in the annexed figure.
The two Northern sides RO, OF, constitute the roof,
and for the most part have no doors; on the East is
a small door for the Women; on the West a much larger
one for the Men; the South side or floor is usually doorless.
Square and triangular houses are not allowed,
and for this reason. The angles of a Square
(and still more those of an equilateral Triangle,)
being much more pointed than those of a Pentagon,
and the lines of inanimate objects (such as houses)
being dimmer than the lines of Men and Women,
it follows that there is no little danger
lest the points of a square of triangular house
residence might do serious injury to an inconsiderate
or perhaps absentminded traveller suddenly running against them:
and therefore, as early as the eleventh century of our era,
triangular houses were universally forbidden by Law,
the only exceptions being fortifications, powder-magazines,
barracks, and other state buildings, which is not desirable
that the general public should approach without circumspection.
At this period, square houses were still everywhere permitted,
though discouraged by a special tax. But, about three centuries
afterwards, the Law decided that in all towns containing a population
above ten thousand, the angle of a Pentagon was the smallest
house-angle that could be allowed consistently with the public safety.
The good sense of the community has seconded the efforts of the
Legislature; and now, even in the country, the pentagonal construction
has superseded every other. It is only now and then in some very
remote and backward agricultural district that an antiquarian may
still discover a square house.
SECTION 3 Concerning the Inhabitants of Flatland
The greatest length or breadth of a full grown inhabitant
of Flatland may be estimated at about eleven of your inches.
Twelve inches may be regarded as a maximum.
Our Women are Straight Lines.
Our Soldiers and Lowest Class of Workmen are Triangles
with two equal sides, each about eleven inches long,
and a base or third side so short (often not exceeding
half an inch) that they form at their vertices
a very sharp and formidable angle. Indeed when their bases
are of the most degraded type (not more than the eighth part
of an inch in size), they can hardly be distinguished from
Straight lines or Women; so extremely pointed are their vertices.
With us, as with you, these Triangles are distinguished
from others by being called Isosceles; and by this name
I shall refer to them in the following pages.
Our Middle Class consists of Equilateral or Equal-Sided Triangles.
Our Professional Men and Gentlemen are Squares (to which class
I myself belong) and Five-Sided Figures or Pentagons.
Next above these come the Nobility, of whom there
are several degrees, beginning at Six-Sided Figures,
or Hexagons, and from thence rising in the number of their
sides till they receive the honourable title of Polygonal,
or many-Sided. Finally when the number of the sides becomes so numerous,
and the sides themselves so small, that the figure cannot be
distinguished from a circle, he is included in the Circular
or Priestly order; and this is the highest class of all.
It is a Law of Nature with us that a male child shall have
one more side than his father, so that each generation
shall rise (as a rule) one step in the scale of development
and nobility. Thus the son of a Square is a Pentagon;
the son of a Pentagon, a Hexagon; and so on.
But this rule applies not always to the Tradesman,
and still less often to the Soldiers, and to the Workmen;
who indeed can hardly be said to deserve the name of human Figures,
since they have not all their sides equal. With them therefore
the Law of Nature does not hold; and the son of an Isosceles
(i.e. a Triangle with two sides equal) remains Isosceles still.
Nevertheless, all hope is not such out, even from the Isosceles,
that his posterity may ultimately rise above his degraded condition.
For, after a long series of military successes, or diligent
and skillful labours, it is generally found that the more
intelligent among the Artisan and Soldier classes manifest
a slight increase of their third side or base, and a shrinkage
of the two other sides. Intermarriages (arranged by the Priests)
between the sons and daughters of these more intellectual
members of the lower classes generally result in an offspring
approximating still more to the type of the Equal-Sided Triangle.
Rarely--in proportion to the vast numbers of Isosceles births--
is a genuine and certifiable Equal-Sided Triangle produced from
Isosceles parents (footnote 1). Such a birth requires, as its
antecedents, not only a series of carefully arranged intermarriages,
but also a long-continued exercise of frugality and self-control
on the part of the would-be ancestors of the coming Equilateral,
and a patient, systematic, and continuous development of the Isosceles
intellect through many generations.
The birth of a True Equilateral Triangle from Isosceles parents
is the subject of rejoicing in our country for many furlongs round.
After a strict examination conducted by the Sanitary and Social Board,
the infant, if certified as Regular, is with solemn ceremonial
admitted into the class of Equilaterals. He is then immediately
taken from his proud yet sorrowing parents and adopted by some
childless Equilateral, who is bound by oath never to permit
the child henceforth to enter his former home or so much
as to look upon his relations again, for fear lest the freshly
developed organism may, by force of unconscious imitation,
fall back again into his hereditary level.
The occasional emergence of an Equilateral from the ranks
of his serf-born ancestors is welcomed, not only by the poor
serfs themselves, as a gleam of light and hope shed upon
the monotonous squalor of their existence, but also by
the Aristocracy at large; for all the higher classes
are well aware that these rare phenomena, while they
do little or nothing to vulgarize their own privileges,
serve as almost useful barrier against revolution from below.
Had the acute-angled rabble been all, without exception,
absolutely destitute of hope and of ambition, they might
have found leaders in some of their many seditious outbreaks,
so able as to render their superior numbers and strength
too much even for the wisdom of the Circles.
But a wise ordinance of Nature has decreed that
in proportion as the working-classes increase in intelligence,
knowledge, and all virtue, in that same proportion their
acute angle (which makes them physically terrible)
shall increase also and approximate to their
comparatively harmless angle of the Equilateral Triangle.
Thus, in the most brutal and formidable off the soldier class--
creatures almost on a level with women in their lack of intelligence--
it is found that, as they wax in the mental ability necessary
to employ their tremendous penetrating power to advantage,
so do they wane in the power of penetration itself.
How admirable is the Law of Compensation! And how perfect
a proof of the natural fitness and, I may almost say,
the divine origin of the aristocratic constitution
of the States of Flatland! By a judicious use of this
Law of Nature, the Polygons and Circles are almost always
able to stifle sedition in its very cradle, taking advantage
of the irrepressible and boundless hopefulness of the human mind.
Art also comes to the aid of Law and Order. It is generally
found possible--by a little artificial compression or expansion
on the part of the State physicians--to make some of the more
intelligent leaders of a rebellion perfectly Regular,
and to admit them at once into the privileged classes;
a much larger number, who are still below the standard,
allured by the prospect of being ultimately ennobled,
are induced to enter the State Hospitals, where they
are kept in honourable confinement for life;
one or two alone of the most obstinate, foolish,
and hopelessly irregular are led to execution.
Then the wretched rabble of the Isosceles, planless
and leaderless, are either transfixed without resistance
by the small body of their brethren whom the Chief Circle
keeps in pay for emergencies of this kind; or else more often,
by means of jealousies and suspicious skillfully fomented
among them by the Circular party, they are stirred to mutual warfare,
and perish by one another's angles. No less than one hundred
and twenty rebellions are recorded in our annals, besides minor
outbreaks numbered at two hundred and thirty-five;
and they have all ended thus.
"What need of a certificate?" a Spaceland critic may ask:
"Is not the procreation of a Square Son a certificate
from Nature herself, proving the Equal-sidedness of the Father?"
I reply that no Lady of any position will mary an uncertified Triangle.
Square offspring has sometimes resulted from a slightly Irregular Triangle;
but in almost every such case the Irregularity of the first generation
is visited on the third; which either fails to attain the Pentagonal rank,
or relapses to the Triangular.
SECTION 4 Concerning the Women
If our highly pointed Triangles of the Soldier class are formidable,
it may be readily inferred that far more formidable are our Women.
For, if a Soldier is a wedge, a Woman is a needle; being, so to speak,
ALL point, at least at the two extremities. Add to this the power
of making herself practically invisible at will, and you will perceive
that a Female, in Flatland, is a creature by no means to be trifled with.
But here, perhaps, some of my younger Readers may ask HOW a woman
in Flatland can make herself invisible. This ought, I think,
to be apparent without any explanation. However, a few words
will make it clear to the most unreflecting.
Place a needle on the table. Then, with your eye on the level of
the table, look at it side-ways, and you see the whole length of it;
but look at it end-ways, and you see nothing but a point,
it has become practically invisible. Just so is it with one of our Women.
When her side is turned towards us, we see her as a straight line;
when the end containing her eye or mouth--for with us these
two organs are identical--is the part that meets our eye,
then we see nothing but a highly lustrous point;
but when the back is presented to our view,
then--being only sub-lustrous, and, indeed,
almost as dim as an inanimate object--her hinder
extremity serves her as a kind of Invisible Cap.
The dangers to which we are exposed from our Women must
now be manifest to the meanest capacity of Spaceland.
If even the angle of a respectable Triangle in the
middle class is not without its dangers;
if to run against a Working Man involves a gash;
if collision with an Officer of the military class
necessitates a serious wound; if a mere touch from
the vertex of a Private Soldier brings with it danger of death;
--what can it be to run against a woman, except absolute
and immediate destruction? And when a Woman is invisible,
or visible only as a dim sub-lustrous point,
how difficult must it be, even for the most cautious,
always to avoid collision!
Many are the enactments made at different times in the different
States of Flatland, in order to minimize this peril;
and in the Southern and less temperate climates,
where the force of gravitation is greater,
and human beings more liable to casual
and involuntary motions, the Laws concerning
Women are naturally much more stringent.
But a general view of the Code may be obtained
from the following summary:--
1. Every house shall have one entrance on the Eastern side,
for the use of Females only; by which all females shall enter
"in a becoming and respectful manner" (footnote 1) and not
by the Men's or Western door.
2. No Female shall walk in any public place without continually
keeping up her Peace-cry, under penalty of death.
3. Any Female, duly certified to be suffering from
St. Vitus's Dance, fits, chronic cold accompanied
by violent sneezing, or any disease necessitating
involuntary motions, shall be instantly destroyed.
In some of the States there is an additional Law
forbidding Females, under penalty of death,
from walking or standing in any public place
without moving their backs constantly from
right to left so as to indicate their presence
to those behind them; other oblige a Woman,
when travelling, to be followed by one of her sons,
or servants, or by her husband; others confine
Women altogether in their houses except during
the religious festivals. But it has been found
by the wisest of our Circles or Statesmen
that the multiplication of restrictions on Females
tends not only to the debilitation and diminution
of the race, but also to the increase of domestic
murders to such an extent that a State loses
more than it gains by a too prohibitive Code.
For whenever the temper of the Women is thus exasperated
by confinement at home or hampering regulations abroad,
they are apt to vent their spleen upon their husbands and children;
and in the less temperate climates the whole male population
of a village has been sometimes destroyed in one or two hours
of a simultaneous female outbreak. Hence the Three Laws,
mentioned above, suffice for the better regulated States,
and may be accepted as a rough exemplification of our Female Code.
After all, our principal safeguard is found, not in Legislature,
but in the interests of the Women themselves. For, although they
can inflict instantaneous death by a retrograde movement,
yet unless they can at once disengage their stinging extremity
from the struggling body of their victim, their own frail bodies
are liable to be shattered.
The power of Fashion is also on our side. I pointed out that
in some less civilized States no female is suffered to stand
in any public place without swaying her back from right to left.
This practice has been universal among ladies of any pretensions
to breeding in all well-governed States, as far back as the memory
of Figures can reach. It is considered a disgrace to any state
that legislation should have to enforce what ought to be,
and is in every respectable female, a natural instinct.
The rhythmical and, if I may so say, well-modulated undulation
of the back in our ladies of Circular rank is envied and imitated
by the wife of a common Equilateral, who can achieve nothing beyond
a mere monotonous swing, like the ticking of a pendulum;
and the regular tick of the Equilateral is no less admired
and copied by the wife of the progressive and aspiring Isosceles,
in the females of whose family no "back-motion" of any kind
has become as yet a necessity of life. Hence, in every family
of position and consideration, "back motion" is as prevalent
as time itself; and the husbands and sons in these households
enjoy immunity at least from invisible attacks.
Not that it must be for a moment supposed that our Women are
destitute of affection. But unfortunately the passion of the
moment predominates, in the Frail Sex, over every other consideration.
This is, of course, a necessity arising from their unfortunate
conformation. For as they have no pretensions to an angle,
being inferior in this respect to the very lowest of the Isosceles,
they are consequently wholly devoid of brainpower, and have
neither reflection, judgment nor forethought, and hardly any memory.
Hence, in their fits of fury, they remember no claims and recognize
no distinctions. I have actually known a case where a Woman
has exterminated her whole household, and half an hour afterwards,
when her rage was over and the fragments swept away,
has asked what has become of her husband and children.
Obviously then a Woman is not to be irritated as long as she
is in a position where she can turn round. When you have them
in their apartments--which are constructed with a view
to denying them that power--you can say and do what you like;
for they are then wholly impotent for mischief, and will
not remember a few minutes hence the incident for which
they may be at this moment threatening you with death,
nor the promises which you may have found it necessary
to make in order to pacify their fury.
On the whole we got on pretty smoothly in our domestic relations,
except in the lower strata of the Military Classes. There the want
of tact and discretion on the part of the husbands produces at times
indescribable disasters. Relying too much on the offensive weapons
of their acute angles instead of the defensive organs of good sense
and seasonable simulations, these reckless creatures too often neglect
the prescribed construction of the women's apartments, or irritate
their wives by ill-advised expressions out of doors, which they
refuse immediately to retract. Moreover a blunt and stolid regard
for literal truth indisposes them to make those lavish promises
by which the more judicious Circle can in a moment pacify his consort.
The result is massacre; not, however, without its advantages,
as it eliminates the more brutal and troublesome of the Isosceles;
and by many of our Circles the destructiveness of the Thinner Sex
is regarded as one among many providential arrangements for suppressing
redundant population, and nipping Revolution in the bud.
Yet even in our best regulated and most approximately Circular
families I cannot say that the ideal of family life is so high
as with you in Spaceland. There is peace, in so far as the absence
of slaughter may be called by that name, but there is necessarily
little harmony of tastes or pursuits; and the cautious wisdom
of the Circles has ensured safety at the cost of domestic comfort.
In every Circular or Polygonal household it has been a habit
from time immemorial--and now has become a kind of instinct
among the women of our higher classes--that the mothers and daughters
should constantly keep their eyes and mouths towards their husband
and his male friends; and for a lady in a family of distinction
to turn her back upon her husband would be regarded as a kind of portent,
involving loss of STATUS. But, as I shall soon shew, this custom,
though it has the advantage of safety, is not without disadvantages.
In the house of the Working Man or respectable Tradesman--where the
wife is allowed to turn her back upon her husband, while pursuing
her household avocations--there are at least intervals of quiet,
when the wife is neither seen nor heard, except for the humming sound
of the continuous Peace-cry; but in the homes of the upper classes
there is too often no peace. There the voluble mouth and bright
penetrating eye are ever directed toward the Master of the household;
and light itself is not more persistent than the stream of Feminine
discourse. The tact and skill which suffice to avert a Woman's sting
are unequal to the task of stopping a Woman's mouth; and as the wife
has absolutely nothing to say, and absolutely no constraint of wit,
sense, or conscience to prevent her from saying it, not a few cynics
have been found to aver that they prefer the danger of the death-dealing
but inaudible sting to the safe sonorousness of a Woman's other end.
To my readers in Spaceland the condition of our Women may seen
truly deplorable, and so indeed it is. A Male of the lowest type
of the Isosceles may look forward to some improvement of his angle,
and to the ultimate elevation of the whole of his degraded caste;
but no Woman can entertain such hopes for her sex. "Once a Woman,
always a Woman" is a Decree of Nature; and the very Laws of Evolution
seem suspended in her disfavour. Yet at least we can admire the wise
Prearrangement which has ordained that, as they have no hopes, so they
shall have no memory to recall, and no forethought to anticipate,
the miseries and humiliations which are at once a necessity of their
existence and the basis of the constitution of Flatland.
SECTION 5 Of our Methods of Recognizing one another
You, who are blessed with shade as well as light, you, who are
gifted with two eyes, endowed with a knowledge of perspective,
and charmed with the enjoyment of various colours, you, who can
actually SEE an angle, and contemplate the complete circumference
of a Circle in the happy region of the Three Dimensions--
how shall I make it clear to you the extreme difficulty which we
in Flatland experience in recognizing one another's configuration?
Recall what I told you above. All beings in Flatland, animate and
inanimate, no matter what their form, present TO OUR VIEW the same,
or nearly the same, appearance, viz. that of a straight Line. How then
can one be distinguished from another, where all appear the same?
The answer is threefold. The first means of recognition is the sense
of hearing; which with us is far more highly developed than with you,
and which enables us not only to distinguish by the voice of our
personal friends, but even to discriminate between different classes,
at least so far as concerns the three lowest orders, the Equilateral,
the Square, and the Pentagon--for the Isosceles I take no account.
But as we ascend the social scale, the process of discriminating
and being discriminated by hearing increases in difficulty, partly because
voices are assimilated, partly because the faculty of voice-discrimination
is a plebeian virtue not much developed among the Aristocracy. And wherever
there is any danger of imposture we cannot trust to this method.
Amongst our lowest orders, the vocal organs are developed to a degree
more than correspondent with those of hearing, so that an Isosceles
can easily feign the voice of a Polygon, and, with some training,
that of a Circle himself. A second method is therefore more
commonly resorted to.
FEELING is, among our Women and lower classes--about our upper
classes I shall speak presently--the principal test of recognition,
at all events between strangers, and when the question is, not as to
the individual, but as to the class. What therefore "introduction"
is among the higher classes in Spaceland, that the process of "feeling"
is with us. "Permit me to ask you to feel and be felt by my friend
Mr. So-and-so"--is still, among the more old-fashioned of our
country gentlemen in districts remote from towns, the customary
formula for a Flatland introduction. But in the towns, and among men
of business, the words "be felt by" are omitted and the sentence is
abbreviated to, "Let me ask you to feel Mr. So-and-so"; although it is
assumed, of course, that the "feeling" is to be reciprocal. Among our
still more modern and dashing young gentlemen--who are extremely
averse to superfluous effort and supremely indifferent to the
purity of their native language--the formula is still further
curtailed by the use of "to feel" in a technical sense, meaning,
"to recommend-for- the-purposes-of-feeling-and-being-felt";
and at this moment the "slang" of polite or fast society
in the upper classes sanctions such a barbarism as "Mr. Smith,
permit me to feel Mr. Jones."
Let not my Reader however suppose that "feeling" is with us
the tedious process that it would be with you, or that we find
it necessary to feel right round all the sides of every individual
before we determine the class to which he belongs. Long practice
and training, begun in the schools and continued in the experience
of daily life, enable us to discriminate at once by the sense of touch,
between the angles of an equal-sided Triangle, Square, and Pentagon;
and I need not say that the brainless vertex of an acute-angled
Isosceles is obvious to the dullest touch. It is therefore
not necessary, as a rule, to do more than feel a single angle
of an individual; and this, once ascertained, tells us the class
of the person whom we are addressing, unless indeed he belongs to the
higher sections of the nobility. There the difficulty is much greater.
Even a Master of Arts in our University of Wentbridge has been known
to confuse a ten-sided with a twelve-sided Polygon; and there is hardly
a Doctor of Science in or out of that famous University who could
pretend to decide promptly and unhesitatingly between a twenty-sided
and a twenty-four sided member of the Aristocracy.
Those of my readers who recall the extracts I gave above from the
Legislative code concerning Women, will readily perceive that the
process of introduction by contact requires some care and discretion.
Otherwise the angles might inflict on the unwary Feeling irreparable
injury. It is essential for the safety of the Feeler that the Felt
should stand perfectly still. A start, a fidgety shifting
of the position, yes, even a violent sneeze, has been known before
now to prove fatal to the incautious, and to nip in the bud many
a promising friendship. Especially is this true among the lower classes
of the Triangles. With them, the eye is situated so far from their vertex
that they can scarcely take cognizance of what goes on at that extremity
of their frame. They are, moreover, of a rough coarse nature, not sensitive
to the delicate touch of the highly organized Polygon. What wonder then
if an involuntary toss of the head has ere now deprived the State
of a valuable life!
I have heard that my excellent Grandfather--one of the least
irregular of his unhappy Isosceles class, who indeed obtained,
shortly before his decease, four out of seven votes from the Sanitary
and Social Board for passing him into the class of the Equal-sided--
often deplored, with a tear in his venerable eye, a miscarriage of
this kind, which had occurred to his great-great-great-Grandfather,
a respectable Working Man with an angle or brain of 59 degrees
30 minutes. According to his account, my unfortunately Ancestor,
being afflicted with rheumatism, and in the act of being felt
by a Polygon, by one sudden start accidentally transfixed
the Great Man through the diagonal and thereby, partly in consequence
of his long imprisonment and degradation, and partly because
of the moral shock which pervaded the whole of my Ancestor's relations,
threw back our family a degree and a half in their ascent towards
better things. The result was that in the next generation
the family brain was registered at only 58 degrees,
and not till the lapse of five generations was the lost
ground recovered, the full 60 degrees attained,
and the Ascent from the Isosceles finally achieved.
And all this series of calamities from one little accident
in the process of Feeling.
As this point I think I hear some of my better educated readers exclaim,
"How could you in Flatland know anything about angles and degrees,
or minutes? We SEE an angle, because we, in the region of Space,
can see two straight lines inclined to one another; but you,
who can see nothing but on straight line at a time, or at all events
only a number of bits of straight lines all in one straight line,--
how can you ever discern an angle, and much less register angles
of different sizes?"
I answer that though we cannot SEE angles, we can INFER them, and this
with great precision. Our sense of touch, stimulated by necessity,
and developed by long training, enables us to distinguish angles
far more accurately than your sense of sight, when unaided by a rule
or measure of angles. Nor must I omit to explain that we have great
natural helps. It is with us a Law of Nature that the brain
of the Isosceles class shall begin at half a degree,
or thirty minutes, and shall increase (if it increases at all)
by half a degree in every generation until the goal of 60 degrees
is reached, when the condition of serfdom is quitted, and the freeman
enters the class of Regulars.
Consequently, Nature herself supplies us with an ascending scale
or Alphabet of angles for half a degree up to 60 degrees, Specimen
of which are placed in every Elementary School throughout the land.
Owing to occasional retrogressions, to still more frequent moral
and intellectual stagnation, and to the extraordinary fecundity
of the Criminal and Vagabond classes, there is always a vast superfluity
of individuals of the half degree and single degree class, and a fair
abundance of Specimens up to 10 degrees. These are absolutely
destitute of civil rights; and a great number of them, not having
even intelligence enough for the purposes of warfare, are devoted
by the States to the service of education. Fettered immovably
so as to remove all possibility of danger, they are placed
in the classrooms of our Infant Schools, and there they are utilized
by the Board of Education for the purpose of imparting to the offspring
of the Middle Classes the tact and intelligence which these wretched
creatures themselves are utterly devoid.
In some States the Specimens are occasionally fed and suffered
to exist for several years; but in the more temperate and better
regulated regions, it is found in the long run more advantageous
for the educational interests of the young, to dispense with food,
and to renew the Specimens every month--which is about the average
duration of the foodless existence of the Criminal class.
In the cheaper schools, what is gained by the longer existence
of the Specimen is lost, partly in the expenditure for food,
and partly in the diminished accuracy of the angles, which
are impaired after a few weeks of constant "feeling."
Nor must we forget to add, in enumerating the advantages
of the more expensive system, that it tends, though slightly
yet perceptibly, to the diminution of the redundant Isosceles population--
an object which every statesman in Flatland constantly keeps in view.
On the whole therefore--although I am not ignorant that,
in many popularly elected School Boards, there is a reaction
in favour of "the cheap system" as it is called--
I am myself disposed to think that this is one
of the many cases in which expense is the truest economy.
But I must not allow questions of School Board politics to divert
me from my subject. Enough has been said, I trust, to shew that
Recognition by Feeling is not so tedious or indecisive a process
as might have been supposed; and it is obviously more trustworthy
than Recognition by hearing. Still there remains, as has been pointed
out above, the objection that this method is not without danger.
For this reason many in the Middle and Lower classes, and all
without exception in the Polygonal and Circular orders,
prefer a third method, the description of which
shall be reserved for the next section.
SECTION 6 Of Recognition by Sight
I am about to appear very inconsistent. In the previous sections
I have said that all figures in Flatland present the appearance
of a straight line; and it was added or implied, that it is consequently
impossible to distinguish by the visual organ between individuals
of different classes: yet now I am about to explain to my Spaceland
critics how we are able to recognize one another by the sense of sight.
If however the Reader will take the trouble to refer to the passage
in which Recognition by Feeling is stated to be universal,
he will find this qualification--"among the lower classes."
It is only among the higher classes and in our more temperate
climates that Sight Recognition is practised.
That this power exists in any regions and for any classes is the result
of Fog; which prevails during the greater part of the year in all parts
save the torrid zones. That which is with you in Spaceland an unmixed evil,
blotting out the landscape, depressing the spirits, and enfeebling the health,
is by us recognized as a blessing scarcely inferior to air itself, and as the
Nurse of arts and Parent of sciences. But let me explain my meaning,
without further eulogies on this beneficent Element.
If Fog were non-existent, all lines would appear equally
and indistinguishably clear; and this is actually the case
in those unhappy countries in which the atmosphere is perfectly
dry and transparent. But wherever there is a rich supply of Fog,
objects that are at a distance, say of three feet, are appreciably
dimmer than those at the distance of two feet eleven inches; and the
result is that by careful and constant experimental observation
of comparative dimness and clearness, we are enabled to infer
with great exactness the configuration of the object observed.
An instance will do more than a volume of generalities to make
my meaning clear.
Suppose I see two individuals approaching whose rank I wish to ascertain.
They are, we will suppose, a Merchant and a Physician, or in other words,
an Equilateral Triangle and a Pentagon; how am I to distinguish them?
It will be obvious, to every child in Spaceland who has touched
the threshold of Geometrical Studies, that, if I can bring my eye
so that its glance may bisect an angle (A) of the approaching stranger,
my view will lie as it were evenly between the two sides that are next
to me (viz. CA and AB), so that I shall contemplate the two impartially,
and both will appear of the same size.
Now in the case of (1) the Merchant, what shall I see? I shall see
a straight line DAE, in which the middle point (A) will be very bright
because it is nearest to me; but on either side the line will shade
away RAPIDLY TO DIMNESS, because the sides AC and AB RECEDE RAPIDLY
INTO THE FOG and what appear to me as the Merchant's extremities,
viz. D and E, will be VERY DIM INDEED.
On the other hand in the case of (2) the Physician, though I shall
here also see a line (D'A'E') with a bright centre (A'), yet it will
shade away LESS RAPIDLY to dimness, because the sides (A'C', A'B')
RECEDE LESS RAPIDLY INTO THE FOG: and what appear to me the Physician's
extremities, viz. D' and E', will not be NOT SO DIM as the extremities
of the Merchant.
The Reader will probably understand from these two instances how
--after a very long training supplemented by constant experience--
it is possible for the well-educated classes among us to discriminate
with fair accuracy between the middle and lowest orders, by the sense
of sight. If my Spaceland Patrons have grasped this general conception,
so far as to conceive the possibility of it and not to reject my account
as altogether incredible--I shall have attained all I can reasonably expect.
Were I to attempt further details I should only perplex. Yet for the sake
of the young and inexperienced, who may perchance infer--from the two simple
instances I have given above, of the manner in which I should recognize
my Father and my Sons--that Recognition by sight is an easy affair,
it may be needful to point out that in actual life most of the problems
of Sight Recognition are far more subtle and complex.
If for example, when my Father, the Triangle, approaches me,
he happens to present his side to me instead of his angle, then,
until I have asked him to rotate, or until I have edged my eye around him,
I am for the moment doubtful whether he may not be a Straight Line, or,
in other words, a Woman. Again, when I am in the company of one of my
two hexagonal Grandsons, contemplating one of his sides (AB) full front,
it will be evident from the accompanying diagram that I shall see one whole
line (AB) in comparative brightness (shading off hardly at all at the ends)
and two smaller lines (CA and BD) dim throughout and shading away into greater
dimness towards the extremities C and D.
But I must not give way to the temptation of enlarging on these topics.
The meanest mathematician in Spaceland will readily believe me when
I assert that the problems of life, which present themselves to the
well-educated--when they are themselves in motion, rotating,
advancing or retreating, and at the same time attempting
to discriminate by the sense of sight between a number of Polygons
of high rank moving in different directions, as for example in a
ball-room or conversazione--must be of a nature to task the angularity
of the most intellectual, and amply justify the rich endowments
of the Learned Professors of Geometry, both Static and Kinetic,
in the illustrious University of Wentbridge, where the Science
and Art of Sight Recognition are regularly taught to large classes
of the ELITE of the States.
It is only a few of the scions of our noblest and wealthiest houses,
who are able to give the time and money necessary for the thorough
prosecution of this noble and valuable Art. Even to me, a Mathematician
of no mean standing, and the Grandfather of two most hopeful and perfectly
regular Hexagons, to find myself in the midst of a crowd of rotating Polygons
of the higher classes, is occasionally very perplexing. And of course
to a common Tradesman, or Serf, such a sight is almost as unintelligible
as it would be to you, my Reader, were you suddenly transported to my country.
In such a crowd you could see on all sides of you nothing but a Line,
apparently straight, but of which the parts would vary irregularly
and perpetually in brightness or dimness. Even if you had completed
your third year in the Pentagonal and Hexagonal classes in the University,
and were perfect in the theory of the subject, you would still find there
was need of many years of experience, before you could move in a fashionable
crowd without jostling against your betters, whom it is against etiquette
to ask to "feel," and who, by their superior culture and breeding,
know all about your movements, while you know very little or nothing
about theirs. In a word, to comport oneself with perfect propriety
in Polygonal society, one ought to be a Polygon oneself.
Such at least is the painful teaching of my experience.
It is astonishing how much the Art--or I may almost call it instinct--
of Sight Recognition is developed by the habitual practice of it
and by the avoidance of the custom of "Feeling." Just as, with you,
the deaf and dumb, if once allowed to gesticulate and to use the hand-alphabet,
will never acquire the more difficult but far more valuable art of lip-speech
and lip-reading, so it is with us as regards "Seeing" and "Feeling."
None who in early life resort to "Feeling" will ever learn "Seeing"
For this reason, among our Higher Classes, "Feeling" is discouraged
or absolutely forbidden. From the cradle their children, instead of going
to the Public Elementary schools (where the art of Feeling is taught,)
are sent to higher Seminaries of an exclusive character; and at our
illustrious University, to "feel" is regarded as a most serious fault,
involving Rustication for the first offence, and Expulsion for the second.
But among the lower classes the art of Sight Recognition is regarded
as an unattainable luxury. A common Tradesman cannot afford to let
his son spend a third of his life in abstract studies. The children
of the poor are therefore allowed to "feel" from their earliest years,
and they gain thereby a precocity and an early vivacity which contrast
at first most favourably with the inert, undeveloped, and listless
behaviour of the half-instructed youths of the Polygonal class;
but when the latter have at last completed their University course,
and are prepared to put their theory into practice, the change
that comes over them may almost be described as a new birth,
and in every art, science, and social pursuit they rapidly
overtake and distance their Triangular competitors.
Only a few of the Polygonal Class fail to pass the Final Test
or Leaving Examination at the University. The condition of the
unsuccessful minority is truly pitiable. Rejected from the higher
class, they are also despised by the lower. They have neither the
matured and systematically trained powers of the Polygonal Bachelors
and Masters of Arts, nor yet the native precocity and mercurial
versatility of the youthful Tradesman. The professions, the public
services, are closed against them, and though in most States they
are not actually debarred from marriage, yet they have the greatest
difficulty in forming suitable alliances, as experience shews that
the offspring of such unfortunate and ill-endowed parents is generally
itself unfortunate, if not positively Irregular.
It is from these specimens of the refuse of our Nobility that the great
Tumults and Seditions of past ages have generally derived their leaders;
and so great is the mischief thence arising that an increasing minority
of our more progressive Statesmen are of opinion that true mercy would
dictate their entire suppression, by enacting that all who fail to pass
the Final Examination of the University should be either imprisoned
for life, or extinguished by a painless death.
But I find myself digressing into the subject of Irregularities,
a matter of such vital interest that it demands a separate section.
SECTION 7 Concerning Irregular Figures
Throughout the previous pages I have been assuming--what perhaps
should have been laid down at the beginning as a distinct and
fundamental proposition--that every human being in Flatland
is a Regular Figure, that is to say of regular construction.
By this I mean that a Woman must not only be a line, but a straight line;
that an Artisan or Soldier must have two of his sides equal;
that Tradesmen must have three sides equal; Lawyers (of which class
I am a humble member), four sides equal, and, generally,
that in every Polygon, all the sides must be equal.
The sizes of the sides would of course depend upon the age of
the individual. A Female at birth would be about an inch long,
while a tall adult Woman might extend to a foot. As to the Males
of every class, it may be roughly said that the length of an adult's size,
when added together, is two feet or a little more. But the size of our sides
is not under consideration. I am speaking of the EQUALITY of sides,
and it does not need much reflection to see that the whole of the
social life in Flatland rests upon the fundamental fact that
Nature wills all Figures to have their sides equal.
If our sides were unequal our angles might be unequal. Instead of
its being sufficient to feel, or estimate by sight, a single angle
in order to determine the form of an individual, it would be necessary
to ascertain each angle by the experiment of Feeling. But life would
be too short for such a tedious groping. The whole science and art
of Sight Recognition would at once perish; Feeling, so far as it is
an art, would not long survive; intercourse would become perilous
or impossible; there would be an end to all confidence, all forethought;
no one would be safe in making the most simple social arrangements;
in a word, civilization might relapse into barbarism.
Am I going too fast to carry my Readers with me to these obvious conclusions?
Surely a moment's reflection, and a single instance from common life,
must convince every one that our social system is based upon Regularity,
or Equality of Angles. You meet, for example, two or three Tradesmen
in the street, whom your recognize at once to be Tradesman by a glance
at their angles and rapidly bedimmed sides, and you ask them to step into
your house to lunch. This you do at present with perfect confidence,
because everyone knows to an inch or two the area occupied by
an adult Triangle: but imagine that your Tradesman drags behind
his regular and respectable vertex, a parallelogram of twelve
or thirteen inches in diagonal:--what are you to do with such
a monster sticking fast in your house door?
But I am insulting the intelligence of my Readers by accumulating
details which must be patent to everyone who enjoys the advantages
of a Residence in Spaceland. Obviously the measurements of a single
angle would no longer be sufficient under such portentous circumstances;
one's whole life would be taken up in feeling or surveying the perimeter
of one's acquaintances. Already the difficulties of avoiding a collision
in a crowd are enough to tax the sagacity of even a well-educated Square;
but if no one could calculate the Regularity of a single figure in the company,
all would be chaos and confusion, and the slightest panic would cause
serious injuries, or--if there happened to be any Women or Soldiers present--
perhaps considerable loss of life.
Expediency therefore concurs with Nature in stamping the seal
of its approval upon Regularity of conformation: nor has the Law
been backward in seconding their efforts. "Irregularity of Figure"
means with us the same as, or more than, a combination of moral obliquity
and criminality with you, and is treated accordingly. There are not
wanting, it is true, some promulgators of paradoxes who maintain that
there is no necessary connection between geometrical and moral Irregularity.
"The Irregular," they say, "is from his birth scouted by his own parents,
derided by his brothers and sisters, neglected by the domestics,
scorned and suspected by society, and excluded from all posts
of responsibility, trust, and useful activity. His every
movement is jealously watched by the police till he comes
of age and presents himself for inspection; then he is either destroyed,
if he is found to exceed the fixed margin of deviation, at an uninteresting
occupation for a miserable stipend; obliged to live and board at the office,
and to take even his vacation under close supervision; what wonder that
human nature, even in the best and purest, is embittered and perverted
by such surroundings!"
All this very plausible reasoning does not convince me, as it has not
convinced the wisest of our Statesmen, that our ancestors erred in laying
it down as an axiom of policy that the toleration of Irregularity
is incompatible with the safety of the State. Doubtless, the life
of an Irregular is hard; but the interests of the Greater Number
require that it shall be hard. If a man with a triangular front
and a polygonal back were allowed to exist and to propagate a still
more Irregular posterity, what would become of the arts of life?
Are the houses and doors and churches in Flatland to be altered
in order to accommodate such monsters? Are our ticket-collectors
to be required to measure every man's perimeter before they allow
him to enter a theatre, or to take his place in a lecture room?
Is an Irregular to be exempted from the militia? And if not,
how is he to be prevented from carrying desolation into the ranks
of his comrades? Again, what irresistible temptations to fraudulent
impostures must needs beset such a creature! How easy for him to enter
a shop with his polygonal front foremost, and to order goods to any extent
from a confiding tradesman! Let the advocates of a falsely called
Philanthropy plead as they may for the abrogation of the Irregular Penal Laws,
I for my part have never known an Irregular who was not also what Nature
evidently intended him to be--a hypocrite, a misanthropist, and,
up to the limits of his power, a perpetrator of all manner of mischief.
Not that I should be disposed to recommend (at present)
the extreme measures adopted by some States, where an infant
whose angle deviates by half a degree from the correct angularity
is summarily destroyed at birth. Some of our highest and ablest men,
men of real genius, have during their earliest days laboured under
deviations as great as, or even greater than forty-five minutes:
and the loss of their precious lives would have been an irreparable
injury to the State. The art of healing also has achieved some
of its most glorious triumphs in the compressions, extensions,
trepannings, colligations, and other surgical or diaetetic
operations by which Irregularity has been partly or wholly cured.
Advocating therefore a VIA MEDIA, I would lay down no fixed
or absolute line of demarcation; but at the period when the frame
is just beginning to set, and when the Medical Board has reported
that recovery is improbably, I would suggest that the Irregular
offspring be painlessly and mercifully consumed.
SECTION 8 Of the Ancient Practice of Painting
If my Readers have followed me with any attention up to this point,
they will not be surprised to hear that life is somewhat dull in Flatland.
I do not, of course, mean that there are not battles, conspiracies,
tumults, factions, and all those other phenomena which are supposed
to make History interesting; nor would I deny that the strange mixture
of the problems of life and the problems of Mathematics, continually
inducing conjecture and giving an opportunity of immediate verification,
imparts to our existence a zest which you in Spaceland can hardly comprehend.
I speak now from the aesthetic and artistic point of view when I say that life
with us is dull; aesthetically and artistically, very dull indeed.
How can it be otherwise, when all one's prospect, all one's landscapes,
historical pieces, portraits, flowers, still life, are nothing but
a single line, with no varieties except degrees of brightness and obscurity?
It was not always thus. Colour, if Tradition speaks the truth,
once for the space of half a dozen centuries or more, threw a transient
splendour over the lives of our ancestors in the remotest ages.
Some private individual--a Pentagon whose name is variously reported--
having casually discovered the constituents of the simpler colours
and a rudimentary method of painting, is said to have begun by decorating
first his house, then his slaves, then his Father, his Sons, and Grandsons,
lastly himself. The convenience as well as the beauty of the results
commended themselves to all. Wherever Chromatistes,--for by that name
the most trustworthy authorities concur in calling him,--turned his
variegated frame, there he at once excited attention, and attracted respect.
No one now needed to "feel" him; no one mistook his front for his back;
all his movements were readily ascertained by his neighbours without
the slightest strain on their powers of calculation; no one jostled him,
or failed to make way for him; his voice was saved the labour
of that exhausting utterance by which we colourless Squares
and Pentagons are often forced to proclaim our individuality
when we move amid a crowd of ignorant Isosceles.
The fashion spread like wildfire. Before a week was over,
every Square and Triangle in the district had copied the example
of Chromatistes, and only a few of the more conservative Pentagons
still held out. A month or two found even the Dodecagons infected
with the innovation. A year had not elapsed before the habit
had spread to all but the very highest of the Nobility.
Needless to say, the custom soon made its way from the district
of Chromatistes to surrounding regions; and within two generations
no one in all Flatland was colourless except the Women and the Priests.
Here Nature herself appeared to erect a barrier, and to plead
against extending the innovations to these two classes. Many-
sidedness was almost essential as a pretext for the Innovators.
"Distinction of sides is intended by Nature to imply distinction
of colours"--such was the sophism which in those days flew from
mouth to mouth, converting whole towns at a time to a new culture.
But manifestly to our Priests and Women this adage did not apply.
The latter had only one side, and therefore--plurally and pedantically
speaking--NO SIDES. The former--if at least they would assert their
claim to be readily and truly Circles, and not mere high-class Polygons,
with an infinitely large number of infinitesimally small sides--
were in the habit of boasting (what Women confessed and deplored)
that they also had no sides, being blessed with a perimeter
of only one line, or, in other words, a Circumference.
Hence it came to pass that these two Classes could see
no force in the so-called axiom about "Distinction of Sides
implying Distinction of Colour;" and when all others
had succumbed to the fascinations of corporal decoration,
the Priests and the Women alone still remained pure
from the pollution of paint.
Immoral, licentious, anarchical, unscientific--call them by what names
you will--yet, from an aesthetic point of view, those ancient days
of the Colour Revolt were the glorious childhood of Art in Flatland--
a childhood, alas, that never ripened into manhood, nor even reached
the blossom of youth. To live then in itself a delight, because living
implied seeing. Even at a small party, the company was a pleasure to behold;
the richly varied hues of the assembly in a church or theatre are said
to have more than once proved too distracting from our greatest teachers
and actors; but most ravishing of all is said to have been the unspeakable
magnificence of a military review.
The sight of a line of battle of twenty thousand Isosceles suddenly
facing about, and exchanging the sombre black of their bases for the
orange of the two sides including their acute angle; the militia
of the Equilateral Triangles tricoloured in red, white, and blue;
the mauve, ultra-marine, gamboge, and burnt umber of the Square
artillerymen rapidly rotating near their vermillion guns;
the dashing and flashing of the five-coloured and six-coloured
Pentagons and Hexagons careering across the field in their
offices of surgeons, geometricians and aides-de-camp--all
these may well have been sufficient to render credible the
famous story how an illustrious Circle, overcome by the artistic
beauty of the forces under his command, threw aside
his marshal's baton and his royal crown, exclaiming
that he henceforth exchanged them for the artist's pencil.
How great and glorious the sensuous development of these days must
have been is in part indicated by the very language and vocabulary
of the period. The commonest utterances of the commonest citizens
in the time of the Colour Revolt seem to have been suffused with a richer
tinge of word or thought; and to that era we are even now indebted
for our finest poetry and for whatever rhythm still remains in the more
scientific utterance of those modern days.
SECTION 9 Of the Universal Colour Bill
But meanwhile the intellectual Arts were fast decaying.
The Art of Sight Recognition, being no longer needed, was
no longer practised; and the studies of Geometry, Statics, Kinetics,
and other kindred subjects, came soon to be considered superfluous,
and feel into disrespect and neglect even at our University.
The inferior Art of Feeling speedily experienced the same fate
at our Elementary Schools. Then the Isosceles classes,
asserting that the Specimens were no longer used nor needed,
and refusing to pay the customary tribute from the Criminal classes
to the service of Education, waxed daily more numerous and more insolent
on the strength of their immunity from the old burden which had formerly
exercised the twofold wholesome effect of at once taming their brutal
nature and thinning their excessive numbers.
Year by year the Soldiers and Artisans began more vehemently
to assert--and with increasing truth--that there was no great
difference between them and the very highest class of Polygons,
now that they were raised to an equality with the latter, and enabled
to grapple with all the difficulties and solve all the problems of life,
whether Statical or Kinetical, by the simple process of Colour Recognition.
Not content with the natural neglect into which Sight Recognition was falling,
they began boldly to demand the legal prohibition of all "monopolizing
and aristocratic Arts" and the consequent abolition of all endowments
for the studies of Sight Recognition, Mathematics, and Feeling.
Soon, they began to insist that inasmuch as Colour, which was
a second Nature, had destroyed the need of aristocratic distinctions,
the Law should follow in the same path, and that henceforth all
individuals and all classes should be recognized as absolutely equal
and entitled to equal rights.
Finding the higher Orders wavering and undecided, the leaders
of the Revolution advanced still further in their requirements,
and at last demanded that all classes alike, the Priests and the Women
not excepted, should do homage to Colour by submitting to be painted.
When it was objected that Priests and Women had no sides, they retorted
that Nature and Expediency concurred in dictating that the front half
of every human being (that is to say, the half containing his eye and mouth)
should be distinguishable from his hinder half. They therefore brought
before a general and extraordinary Assembly of all the States of Flatland
a Bill proposing that in every Woman the half containing the eye and mouth
should be coloured red, and the other half green. The Priests were to be
painted in the same way, red being applied to that semicircle in which
the eye and mouth formed the middle point; while the other or hinder
semicircle was to be coloured green.
There was no little cunning in this proposal, which indeed emanated
not from any Isosceles--for no being so degraded would have angularity
enough to appreciate, much less to devise, such a model of state-craft--
but from an Irregular Circle who, instead of being destroyed in his childhood,
was reserved by a foolish indulgence to bring desolation on his country
and destruction on myriads of followers.
On the one hand the proposition was calculated to bring the Women
in all classes over to the side of the Chromatic Innovation.
For by assigning to the Women the same two colours as were assigned
to the Priests, the Revolutionists thereby ensured that, in certain
positions, every Woman would appear as a Priest, and be treated with
corresponding respect and deference--a prospect that could not fail
to attract the Female Sex in a mass.
But by some of my Readers the possibility of the identical appearance
of Priests and Women, under a new Legislation, may not be recognized;
if so, a word or two will make it obvious.
Imagine a woman duly decorated, according to the new Code; with the
front half (i.e., the half containing the eye and mouth) red,
and with the hinder half green. Look at her from one side.
Obviously you will see a straight line, HALF RED, HALF GREEN.
Now imagine a Priest, whose mouth is at M, and whose front
semicircle (AMB) is consequently coloured red, while his hinder
semicircle is green; so that the diameter AB divides the green
from the red. If you contemplate the Great Man so as to have your eye
in the same straight line as his dividing diameter (AB), what you will
see will be a straight line (CBD), of which ONE HALF (CB) WILL BE RED,
AND THE OTHER (BD) GREEN. The whole line (CD) will be rather shorter
perhaps than that of a full-sized Woman, and will shade off more rapidly
towards its extremities; but the identity of the colours would give you
an immediate impression of identity in Class, making you neglectful
of other details. Bear in mind the decay of Sight Recognition
which threatened society at the time of the Colour revolt;
add too the certainty that Woman would speedily learn to
shade off their extremities so as to imitate the Circles;
it must then be surely obvious to you, my dear Reader,
that the Colour Bill placed us under a great danger
of confounding a Priest with a young Woman.
How attractive this prospect must have been to the Frail Sex may
readily be imagined. They anticipated with delight the confusion that
would ensue. At home they might hear political and ecclesiastical
secrets intended not for them but for their husbands and brothers, and
might even issue some commands in the name of a priestly Circle; out
of doors the striking combination of red and green without addition
of any other colours, would be sure to lead the common people into
endless mistakes, and the Woman would gain whatever the Circles lost,
in the deference of the passers by. As for the scandal that would
befall the Circular Class if the frivolous and unseemly conduct of the
Women were imputed to them, and as to the consequent subversion of the
Constitution, the Female Sex could not be expected to give a thought
to these considerations. Even in the households of the Circles, the
Women were all in favour of the Universal Colour Bill.
The second object aimed at by the Bill was the gradual
demoralization of the Circles themselves. In the general intellectual
decay they still preserved their pristine clearness and strength of
understanding. From their earliest childhood, familiarized in their
Circular households with the total absence of Colour, the Nobles alone
preserved the Sacred Art of Sight Recognition, with all the advantages
that result from that admirable training of the intellect. Hence, up
to the date of the introduction of the Universal Colour Bill, the
Circles had not only held their own, but even increased their lead of
the other classes by abstinence from the popular fashion.
Now therefore the artful Irregular whom I described above as
the real author of this diabolical Bill, determined at one blow
to lower the status of the Hierarchy by forcing them to submit
to the pollution of Colour, and at the same time to destroy their
domestic opportunities of training in the Art of Sight Recognition,
so as to enfeeble their intellects by depriving them of their pure
and colourless homes. Once subjected to the chromatic taint,
every parental and every childish Circle would demoralize each other.
Only in discerning between the Father and the Mother would the Circular
infant find problems for the exercise of his understanding--problems
too often likely to be corrupted by maternal impostures with the
result of shaking the child's faith in all logical conclusions.
Thus by degrees the intellectual lustre of the Priestly Order would wane,
and the road would then lie open for a total destruction of all Aristocratic
Legislature and for the subversion of our Privileged Classes.
SECTION 10 Of the Suppression of the Chromatic Sedition
The agitation for the Universal Colour Bill continued for three years;
and up to the last moment of that period it seemed as though Anarchy
were destined to triumph.
A whole army of Polygons, who turned out to fight as private soldiers,
was utterly annihilated by a superior force of Isosceles Triangles--
the Squares and Pentagons meanwhile remaining neutral.
Worse than all, some of the ablest Circles fell a prey to conjugal fury.
Infuriated by political animosity, the wives in many a noble household
wearied their lords with prayers to give up their opposition to the
Colour Bill; and some, finding their entreaties fruitless, fell on
and slaughtered their innocent children and husband, perishing
themselves in the act of carnage. It is recorded that during
that triennial agitation no less than twenty-three Circles
perished in domestic discord.
Great indeed was the peril. It seemed as though the Priests
had no choice between submission and extermination; when suddenly
the course of events was completely changed by one of those picturesque
incidents which Statesmen ought never to neglect, often to anticipate,
and sometimes perhaps to originate, because of the absurdly
disproportionate power with which they appeal to the sympathies
of the populace.
It happened that an Isosceles of a low type, with a brain little
if at all above four degrees--accidentally dabbling in the colours
of some Tradesman whose shop he had plundered--painted himself,
or caused himself to be painted (for the story varies) with the twelve
colours of a Dodecagon. Going into the Market Place he accosted
in a feigned voice a maiden, the orphan daughter of a noble Polygon,
whose affection in former days he had sought in vain; and by a series
of deceptions--aided, on the one side, by a string of lucky accidents
too long to relate, and, on the other, by an almost inconceivable
fatuity and neglect of ordinary precautions on the part of the
relations of the bride--he succeeded in consummating the marriage.
The unhappy girl committed suicide on discovering the fraud to which
she had been subjected.
When the news of this catastrophe spread from State to State
the minds of the Women were violently agitated. Sympathy with
the miserable victim and anticipations of similar deceptions
for themselves, their sisters, and their daughters, made them now regard
the Colour Bill in an entirely new aspect. Not a few openly avowed
themselves converted to antagonism; the rest needed only a slight
stimulus to make a similar avowal. Seizing this favourable opportunity,
the Circles hastily convened an extraordinary Assembly of the States;
and besides the usual guard of Convicts, they secured the attendance
of a large number of reactionary Women.
Amidst an unprecedented concourse, the Chief Circle of those days
--by name Pantocyclus--arose to find himself hissed and hooted by a
hundred and twenty thousand Isosceles. But he secured silence
by declaring that henceforth the Circles would enter on a policy
of Concession; yielding to the wishes of the majority,
they would accept the Colour Bill. The uproar being
at once converted to applause, he invited Chromatistes,
the leader of the Sedition, into the centre of the hall,
to receive in the name of his followers the submission
of the Hierarchy. Then followed a speech, a masterpiece
of rhetoric, which occupied nearly a day in the delivery,
and to which no summary can do justice.
With a grave appearance of impartiality he declared that as they
were now finally committing themselves to Reform or Innovation,
it was desirable that they should take one last view of the perimeter
of the whole subject, its defects as well as its advantages.
Gradually introduction the mention of the dangers to the Tradesmen,
the Professional Classes and the Gentlemen, he silenced the rising murmurs
of the Isosceles by reminding them that, in spite of all these defects,
he was willing to accept the Bill if it was approved by the majority.
But it was manifest that all, except the Isosceles, were moved
by his words and were either neutral or averse to the Bill.
Turning now to the Workmen he asserted that their interests must
not be neglected, and that, if they intended to accept the Colour Bill,
they ought at least to do so with full view of the consequences.
Many of them, he said, were on the point of being admitted
to the class of the Regular Triangles; others anticipated
for their children a distinction they could not hope for themselves.
That honourable ambition would not have to be sacrificed. With the
universal adoption of Colour, all distinctions would cease;
Regularity would be confused with Irregularity;
development would give place to retrogression;
the Workman would in a few generations be degraded
to the level of the Military, or even the Convict Class;
political power would be in the hands of the greatest number,
that is to say the Criminal Classes, who were already more
numerous than the Workmen, and would soon out-number all
the other Classes put together when the usual Compensative Laws
of Nature were violated.
A subdued murmur of assent ran through the ranks of the Artisans,
and Chromatistes, in alarm, attempted to step forward and address them.
But he found himself encompassed with guards and forced to remain silent
while the Chief Circle in a few impassioned words made a final appeal
to the Women, exclaiming that, if the Colour Bill passed, no marriage
would henceforth be safe, no woman's honour secure; fraud, deception,
hypocrisy would pervade every household; domestic bliss would share
the fate of the Constitution and pass to speedy perdition.
"Sooner than this," he cried, "come death."
At these words, which were the preconcerted signal for action,
the Isosceles Convicts fell on and transfixed the wretched Chromatistes;
the Regular Classes, opening their ranks, made way for a band of Women who,
under direction of the Circles, moved back foremost, invisibly and unerringly
upon the unconscious soldiers; the Artisans, imitating the example of their
betters, also opened their ranks. Meantime bands of Convicts occupied
every entrance with an impenetrable phalanx.
The battle, or rather carnage, was of short duration. Under the
skillful generalship of the Circles almost every Woman's charge
was fatal and very many extracted their sting uninjured, ready for
a second slaughter. But no second blow was needed; the rabble
of the Isosceles did the rest of the business for themselves.
Surprised, leader-less, attacked in front by invisible foes,
and finding egress cut off by the Convicts behind them, they at once--
after their manner--lost all presence of mind, and raised the cry
of "treachery." This sealed their fate. Every Isosceles now saw
and felt a foe in every other. In half an hour not one of that vast
multitude was living; and the fragments of seven score thousand
of the Criminal Class slain by one another's angles attested
the triumph of Order.
The Circles delayed not to push their victory to the uttermost.
The Working Men they spared but decimated. The Militia of the
Equilaterals was at once called out, and every Triangle suspected
of Irregularity on reasonable grounds, was destroyed by Court Martial,
without the formality of exact measurement by the Social Board.
The homes of the Military and Artisan classes were inspected in a course
of visitation extending through upwards of a year; and during that period
every town, village, and hamlet was systematically purged of that excess
of the lower orders which had been brought about by the neglect to pay
the tribute of Criminals to the Schools and University, and by the violation
of other natural Laws of the Constitution of Flatland. Thus the balance
of classes was again restored.
Needless to say that henceforth the use of Colour was abolished,
and its possession prohibited. Even the utterance of any word
denoting Colour, except by the Circles or by qualified scientific
teachers, was punished by a severe penalty. Only at our University
in some of the very highest and most esoteric classes--which I myself
have never been privileged to attend--it is understood that the
sparing use of Colour is still sanctioned for the purpose
of illustrating some of the deeper problems of mathematics.
But of this I can only speak from hearsay.
Elsewhere in Flatland, Colour is now non-existent. The art of making
it is known to only one living person, the Chief Circle for the time being;
and by him it is handed down on his death-bed to none but his Successor.
One manufactory alone produces it; and, lest the secret should be betrayed,
the Workmen are annually consumed, and fresh ones introduced. So great
is the terror with which even now our Aristocracy looks back to the
far-distant days of the agitation for the Universal Colour Bill.
SECTION 11 Concerning our Priests
It is high time that I should pass from these brief and discursive
notes about things in Flatland to the central event of this book,
my initiation into the mysteries of Space. THAT is my subject;
all that has gone before is merely preface.
For this reason I must omit many matters of which the explanation
would not, I flatter myself, be without interest for my Readers:
as for example, our method of propelling and stopping ourselves,
although destitute of feet; the means by which we give fixity
to structures of wood, stone, or brick, although of course
we have no hands, nor can we lay foundations as you can,
nor avail ourselves of the lateral pressure of the earth;
the manner in which the rain originates in the intervals
between our various zones, so that the northern regions
do not intercept the moisture falling on the southern;
the nature of our hills and mines, our trees and vegetables,
our seasons and harvests; our Alphabet and method of writing,
adapted to our linear tablets; these and a hundred other details
of our physical existence I must pass over, nor do I mention them
now except to indicate to my readers that their omission proceeds
not from forgetfulness on the part of the author, but from his
regard for the time of the Reader.
Yet before I proceed to my legitimate subject some few final remarks
will no doubt be expected by my Readers upon these pillars and mainstays
of the Constitution of Flatland, the controllers of our conduct
and shapers of our destiny, the objects of universal homage
and almost of adoration: need I say that I mean our Circles or Priests?
When I call them Priests, let me not be understood as meaning
no more than the term denotes with you. With us, our Priests
are Administrators of all Business, Art, and Science; Directors of Trade,
Commerce, Generalship, Architecture, Engineering, Education,
Statesmanship, Legislature, Morality, Theology; doing nothing
themselves, they are the Causes of everything worth doing,
that is done by others.
Although popularly everyone called a Circle is deemed a Circle,
yet among the better educated Classes it is known that no Circle
is really a Circle, but only a Polygon with a very large number
of very small sides. As the number of the sides increases, a Polygon
approximates to a Circle; and, when the number is very great indeed,
say for example three or four hundred, it is extremely difficult
for the most delicate touch to feel any polygonal angles. Let me
say rather it WOULD be difficult: for, as I have shown above,
Recognition by Feeling is unknown among the highest society,
and to FEEL a Circle would be considered a most audacious insult.
This habit of abstention from Feeling in the best society enables
a Circle the more easily to sustain the veil of mystery in which,
from his earliest years, he is wont to enwrap the exact nature of his
Perimeter or Circumference. Three feet being the average Perimeter
it follows that, in a Polygon of three hundred sides each side will
be no more than the hundredth part of a foot in length, or little more
than the tenth part of an inch; and in a Polygon of six or seven hundred
sides the sides are little larger than the diameter of a Spaceland pin-head.
It is always assumed, by courtesy, that the Chief Circle for the time being
has ten thousand sides.
The ascent of the posterity of the Circles in the social scale is
not restricted, as it is among the lower Regular classes, by the Law
of Nature which limits the increase of sides to one in each generation.
If it were so, the number of sides in the Circle would be a mere question
of pedigree and arithmetic, and the four hundred and ninety-seventh
descendant of an Equilateral Triangle would necessarily be a polygon
with five hundred sides. But this is not the case. Nature's Law
prescribes two antagonistic decrees affecting Circular propagation;
first, that as the race climbs higher in the scale of development,
so development shall proceed at an accelerated pace; second,
that in the same proportion, the race shall become less fertile.
Consequently in the home of a Polygon of four or five hundred sides
it is rare to find a son; more than one is never seen. On the other
hand the son of a five-hundred-sided Polygon has been known to possess
five hundred and fifty, or even six hundred sides.
Art also steps in to help the process of higher Evolution.
Our physicians have discovered that the small and tender sides
of an infant Polygon of the higher class can be fractured, and his whole
frame re-set, with such exactness that a Polygon of two or three
hundred sides sometimes--by no means always, for the process
is attended with serious risk--but sometimes overleaps two or three
hundred generations, and as it were double at a stroke, the number
of his progenitors and the nobility of his descent.
Many a promising child is sacrificed in this way. Scarcely one
out of ten survives. Yet so strong is the parental ambition among
those Polygons who are, as it were, on the fringe of the Circular
class, that it is very rare to find the Nobleman of that position
in society, who has neglected to place his first-born in the Circular
Neo-Therapeutic Gymnasium before he has attained the age of a month.
One year determines success or failure. At the end of that time
the child has, in all probability, added one more to the tombstones
that crowd the Neo-Therapeutic Cemetery; but on rare occasional a glad
procession bears back the little one to his exultant parents, no longer
a Polygon, but a Circle, at least by courtesy: and a single instance
of so blessed a result induces multitudes of Polygonal parents to submit
to similar domestic sacrifice, which have a dissimilar issue.
SECTION 12 Of the Doctrine of our Priests
As to the doctrine of the Circles it may briefly be summed up
in a single maxim, "Attend to your Configuration." Whether political,
ecclesiastical, or moral, all their teaching has for its object
the improvement of individual and collective Configuration--with special
reference of course to the Configuration of the Circles, to which all
other objects are subordinated.
It is the merit of the Circles that they have effectually
suppressed those ancient heresies which led men to waste energy
and sympathy in the vain belief that conduct depends upon will, effort,
training, encouragement, praise, or anything else but Configuration.
It was Pantocyclus--the illustrious Circle mentioned above, as the
queller of the Colour Revolt--who first convinced mankind that
Configuration makes the man; that if, for example, you are born
an Isosceles with two uneven sides, you will assuredly go wrong
unless you have them made even--for which purpose you must go
to the Isosceles Hospital; similarly, if you are a Triangle,
or Square, or even a Polygon, born with any Irregularity,
you must be taken to one of the Regular Hospitals to
have your disease cured; otherwise you will end your days
in the State Prison or by the angle of the State Executioner.
All faults or defects, from the slightest misconduct to the
most flagitious crime, Pantocyclus attributed to some deviation
from perfect Regularity in the bodily figure, caused perhaps
(if not congenital) by some collision in a crowd; by neglect
to take exercise, or by taking too much of it; or even by
a sudden change of temperature, resulting in a shrinkage
or expansion in some too susceptible part of the frame.
Therefore, concluded that illustrious Philosopher,
neither good conduct nor bad conduct is a fit subject,
in any sober estimation, for either praise or blame.
For why should you praise, for example, the integrity
of a Square who faithfully defends the interests of his client,
when you ought in reality rather to admire the exact precision
of his right angles? Or again, why blame a lying,
thievish Isosceles, when you ought rather to deplore
the incurable inequality of his sides?
Theoretically, this doctrine is unquestionable; but it
has practical drawbacks. In dealing with an Isosceles,
if a rascal pleads that he cannot help stealing because
of his unevenness, you reply that for that very reason,
because he cannot help being a nuisance to his neighbours,
you, the Magistrate, cannot help sentencing him to be consumed--
and there's an end of the matter. But in little domestic difficulties,
when the penalty of consumption, or death, is out of the question,
this theory of Configuration sometimes comes in awkwardly;
and I must confess that occasionally when one of my own Hexagonal
Grandsons pleads as an excuse for his disobedience that a sudden
change of temperature has been too much for his Perimeter,
and that I ought to lay the blame not on him but on his Configuration,
which can only be strengthened by abundance of the choicest sweetmeats,
I neither see my way logically to reject, nor practically to accept,
For my own part, I find it best to assume that a good sound scolding
or castigation has some latent and strengthening influence on
my Grandson's Configuration; though I own that I have no grounds
for thinking so. At all events I am not alone in my way of extricating
myself from this dilemma; for I find that many of the highest Circles,
sitting as Judges in law courts, use praise and blame towards Regular
and Irregular Figures; and in their homes I know by experience that,
when scolding their children, they speak about "right" and "wrong"
as vehemently and passionately as if they believe that these names
represented real existence, and that a human Figure is really capable
of choosing between them.
Constantly carrying out their policy of making Configuration
the leading idea in every mind, the Circles reverse the nature
of that Commandment which in Spaceland regulates the relations between
parents and children. With you, children are taught to honour their parents;
with us--next to the Circles, who are the chief object of universal homage--
a man is taught to honour his Grandson, if he has one; or, if not, his Son.
By "honour," however, is by no means mean "indulgence," but a reverent
regard for their highest interests: and the Circles teach that the duty
of fathers is to subordinate their own interests to those of posterity,
thereby advancing the welfare of the whole State as well as that
of their own immediate descendants.
The weak point in the system of the Circles--if a humble Square
may venture to speak of anything Circular as containing any element
of weakness--appears to me to be found in their relations with Women.
As it is of the utmost importance for Society that Irregular
births should be discouraged, it follows that no Woman who has any
Irregularities in her ancestry is a fit partner for one who desires
that his posterity should rise by regular degrees in the social scale.
Now the Irregularity of a Male is a matter of measurement; but as
all Women are straight, and therefore visibly Regular so to speak, one
has to device some other means of ascertaining what I may call their
invisible Irregularity, that is to say their potential Irregularities
as regards possible offspring. This is effected by carefully-kept
pedigrees, which are preserved and supervised by the State; and without
a certified pedigree no Woman is allowed to marry.
Now it might have been supposed the a Circle--proud of his ancestry
and regardful for a posterity which might possibly issue hereafter
in a Chief Circle--would be more careful than any other to choose
a wife who had no blot on her escutcheon. But it is not so.
The care in choosing a Regular wife appears to diminish as one rises
in the social scale. Nothing would induce an aspiring Isosceles,
who has hopes of generating an Equilateral Son, to take a wife
who reckoned a single Irregularity among her Ancestors; a Square
or Pentagon, who is confident that his family is steadily on the rise,
does not inquire above the five-hundredth generation; a Hexagon
or Dodecagon is even more careless of the wife's pedigree;
but a Circle has been known deliberately to take a wife
who has had an Irregular Great-Grandfather, and all because
of some slight superiority of lustre, or because of the charms
of a low voice--which, with us, even more than with you,
is thought "an excellent thing in a Woman."
Such ill-judged marriages are, as might be expected, barren, if they
do not result in positive Irregularity or in diminution of sides;
but none of these evils have hitherto provided sufficiently deterrent.
The loss of a few sides in a highly-developed Polygon is not easily
noticed, and is sometimes compensated by a successful operation in
the Neo-Therapeutic Gymnasium, as I have described above; and the Circles
are too much disposed to acquiesce in infecundity as a law of the
superior development. Yet, if this evil be not arrested, the gradual
diminution of the Circular class may soon become more rapid, and the
time may not be far distant when, the race being no longer able to
produce a Chief Circle, the Constitution of Flatland must fall.
One other word of warning suggest itself to me, though I cannot
so easily mention a remedy; and this also refers to our relations
with Women. About three hundred years ago, it was decreed by the
Chief Circle that, since women are deficient in Reason but abundant
in Emotion, they ought no longer to be treated as rational, nor receive
any mental education. The consequence was that they were no longer
taught to read, nor even to master Arithmetic enough to enable them
to count the angles of their husband or children; and hence they sensibly
declined during each generation in intellectual power. And this
system of female non-education or quietism still prevails.
My fear is that, with the best intentions, this policy has been
carried so far as to react injuriously on the Male Sex.
For the consequence is that, as things now are, we Males have to
lead a kind of bi-lingual, and I may almost say bimental, existence.
With Women, we speak of "love," "duty," "right," "wrong," "pity,"
"hope," and other irrational and emotional conceptions, which have
no existence, and the fiction of which has no object except to control
feminine exuberances; but among ourselves, and in our books, we have
an entirely different vocabulary and I may also say, idiom. "Love"
them becomes "the anticipation of benefits"; "duty" becomes "necessity"
or "fitness"; and other words are correspondingly transmuted.
Moreover, among Women, we use language implying the utmost deference
for their Sex; and they fully believe that the Chief Circle Himself
is not more devoutly adored by us than they are: but behind their
backs they are both regarded and spoken of--by all but the very young--
as being little better than "mindless organisms."
Our Theology also in the Women's chambers is entirely different
from our Theology elsewhere.
Now my humble fear is that this double training, in language
as well as in thought, imposes somewhat too heavy a burden upon
the young, especially when, at the age of three years old, they are
taken from the maternal care and taught to unlearn the old language--
except for the purpose of repeating it in the presence of the Mothers
and Nurses--and to learn the vocabulary and idiom of science.
Already methinks I discern a weakness in the grasp of mathematical
truth at the present time as compared with the more robust intellect
of our ancestors three hundred years ago. I say nothing of the possible
danger if a Woman should ever surreptitiously learn to read and convey
to her Sex the result of her perusal of a single popular volume;
nor of the possibility that the indiscretion or disobedience of some
infant Male might reveal to a Mother the secrets of the logical dialect.
On the simple ground of the enfeebling of the male intellect,
I rest this humble appeal to the highest Authorities to reconsider
the regulations of Female education.
"O brave new worlds,
That have such people in them!"
SECTION 13 How I had a Vision of Lineland
It was the last day but one of the 1999th year of our era, and the
first day of the Long Vacation. Having amused myself till a late hour
with my favourite recreation of Geometry, I had retired to rest
with an unsolved problem in my mind. In the night I had a dream.
I saw before me a vast multitude of small Straight Lines
(which I naturally assumed to be Women) interspersed with other Beings
still smaller and of the nature of lustrous points--all moving to and fro
in one and the same Straight Line, and, as nearly as I could judge,
with the same velocity.
A noise of confused, multitudinous chirping or twittering issued
from them at intervals as long as they were moving; but sometimes
they ceased from motion, and then all was silence.
Approaching one of the largest of what I thought to be Women,
I accosted her, but received no answer. A second and third appeal
on my part were equally ineffectual. Losing patience at what appeared
to me intolerable rudeness, I brought my mouth to a position full
in front of her mouth so as to intercept her motion, and loudly repeated
my question, "Woman, what signifies this concourse, and this strange
and confused chirping, and this monotonous motion to and fro in one
and the same Straight Line?"
"I am no Woman," replied the small Line: "I am the Monarch of the world.
But thou, whence intrudest thou into my realm of Lineland?"
Receiving this abrupt reply, I begged pardon if I had in any way
startled or molested his Royal Highness; and describing myself
as a stranger I besought the King to give me some account of his dominions.
But I had the greatest possible difficulty in obtaining any information
on points that really interested me; for the Monarch could not refrain
from constantly assuming that whatever was familiar to him must also
be known to me and that I was simulating ignorance in jest.
However, by preserving questions I elicited the following facts:
It seemed that this poor ignorant Monarch--as he called himself--
was persuaded that the Straight Line which he called his Kingdom,
and in which he passed his existence, constituted the whole of the world,
and indeed the whole of Space. Not being able either to move or to see,
save in his Straight Line, he had no conception of anything out of it.
Though he had heard my voice when I first addressed him, the sounds
had come to him in a manner so contrary to his experience that he had
made no answer, "seeing no man," as he expressed it, "and hearing
a voice as it were from my own intestines." Until the moment when
I placed my mouth in his World, he had neither seen me, nor heard
anything except confused sounds beating against, what I called his side,
but what he called his INSIDE or STOMACH; nor had he even now the least
conception of the region from which I had come. Outside his World,
or Line, all was a blank to him; nay, not even a blank, for a blank
implies Space; say, rather, all was non-existent.
His subjects--of whom the small Lines were men and the Points Women--
were all alike confined in motion and eyesight to that single Straight Line,
which was their World. It need scarcely be added that the whole of their
horizon was limited to a Point; nor could any one ever see anything
but a Point. Man, woman, child, thing--each as a Point to the eye
of a Linelander. Only by the sound of the voice could sex or age
be distinguished. Moreover, as each individual occupied the whole
of the narrow path, so to speak, which constituted his Universe,
and no one could move to the right or left to make way for passers by,
it followed that no Linelander could ever pass another. Once neighbours,
always neighbours. Neighbourhood with them was like marriage with us.
Neighbours remained neighbours till death did them part.
Such a life, with all vision limited to a Point, and all motion
to a Straight Line, seemed to me inexpressibly dreary; and I was
surprised to note that vivacity and cheerfulness of the King.
Wondering whether it was possible, amid circumstances so unfavourable
to domestic relations, to enjoy the pleasures of conjugal union,
I hesitated for some time to question his Royal Highness on so delicate
a subject; but at last I plunged into it by abruptly inquiring
as to the health of his family. "My wives and children," he replied,
"are well and happy."
Staggered at this answer--for in the immediate proximity of the Monarch
(as I had noted in my dream before I entered Lineland) there were none
but Men--I ventured to reply, "Pardon me, but I cannot imagine how your
Royal Highness can at any time either see or approach their Majesties,
when there at least half a dozen intervening individuals, whom you can
neither see through, nor pass by? Is it possible that in Lineland
proximity is not necessary for marriage and for the generation of children?"
"How can you ask so absurd a question?" replied the Monarch. "If it were
indeed as you suggest, the Universe would soon be depopulated. No, no;
neighbourhood is needless for the union of hearts; and the birth
of children is too important a matter to have been allowed to depend
upon such an accident as proximity. You cannot be ignorant of this.
Yet since you are pleased to affect ignorance, I will instruct you
as if you were the veriest baby in Lineland. Know, then, that marriages
are consummated by means of the faculty of sound and the sense of hearing.
"You are of course aware that every Man has two mouths or voices--
as well as two eyes--a bass at one and a tenor at the other of his
extremities. I should not mention this, but that I have been
unable to distinguish your tenor in the course of our conversation."
I replied that I had but one voice, and that I had not been aware
that his Royal Highness had two. "That confirms my impression,"
said the King, "that you are not a Man, but a feminine Monstrosity
with a bass voice, and an utterly uneducated ear. But to continue.
"Nature having herself ordained that every Man should wed two wives--"
"Why two?" asked I. "You carry your affected simplicity too far,"
he cried. "How can there be a completely harmonious union without
the combination of the Four in One, viz. the Bass and Tenor of the Man
and the Soprano and Contralto of the two Women?" "But supposing,"
said I, "that a man should prefer one wife or three?" "It is impossible,"
he said; "it is as inconceivable as that two and one should make five,
or that the human eye should see a Straight Line." I would have
interrupted him; but he proceeded as follows:
"Once in the middle of each week a Law of Nature compels us to move
to and fro with a rhythmic motion of more than usual violence,
which continues for the time you would take to count a hundred and one.
In the midst of this choral dance, at the fifty-first pulsation,
the inhabitants of the Universe pause in full career, and each
individual sends forth his richest, fullest, sweetest strain.
It is in this decisive moment that all our marriages are made.
So exquisite is the adaptation of Bass and Treble, of Tenor to Contralto,
that oftentimes the Loved Ones, though twenty thousand leagues away,
recognize at once the responsive note of their destined Lover; and,
penetrating the paltry obstacles of distance, Love unites the three.
The marriage in that instance consummated results in a threefold Male
and Female offspring which takes its place in Lineland."
"What! Always threefold?" said I. "Must one wife then always have twins?"
"Bass-voice Monstrosity! yes," replied the King. "How else could
the balance of the Sexes be maintained, if two girls were not born
for every boy? Would you ignore the very Alphabet of Nature?"
He ceased, speechless for fury; and some time elapsed before
I could induce him to resume his narrative.
"You will not, of course, suppose that every bachelor among us finds
his mates at the first wooing in this universal Marriage Chorus.
On the contrary, the process is by most of us many times repeated.
Few are the hearts whose happy lot is at once to recognize in each
other's voice the partner intended for them by Providence, and to fly
into a reciprocal and perfectly harmonious embrace. With most of us
the courtship is of long duration. The Wooer's voices may perhaps
accord with one of the future wives, but not with both; or not,
at first, with either; or the Soprano and Contralto may not quite
harmonize. In such cases Nature has provided that every weekly Chorus
shall bring the three Lovers into closer harmony. Each trial of voice,
each fresh discovery of discord, almost imperceptibly induces the less
perfect to modify his or her vocal utterance so as to approximate
to the more perfect. And after many trials and many approximations,
the result is at last achieved. There comes a day at last when,
while the wonted Marriage Chorus goes forth from universal Lineland,
the three far-off Lovers suddenly find themselves in exact harmony,
and, before they are aware, the wedded Triplet is rapt vocally into
a duplicate embrace; and Nature rejoices over one more marriage
and over three more births."
SECTION 14 How I vainly tried to explain the nature of Flatland
Thinking that it was time to bring down the Monarch from his raptures
to the level of common sense, I determined to endeavour to open up
to him some glimpses of the truth, that is to say of the nature
of things in Flatland. So I began thus: "How does your Royal Highness
distinguish the shapes and positions of his subjects? I for my part
noticed by the sense of sight, before I entered your Kingdom,
that some of your people are lines and others Points;
and that some of the lines are larger --" "You speak of an impossibility,"
interrupted the King; "you must have seen a vision; for to detect the difference
between a Line and a Point by the sense of sight is, as every one knows,
in the nature of things, impossible; but it can be detected by the sense
of hearing, and by the same means my shape can be exactly ascertained.
Behold me--I am a Line, the longest in Lineland, over six inches of Space --"
"Of Length," I ventured to suggest. "Fool," said he, "Space is Length.
Interrupt me again, and I have done."
I apologized; but he continued scornfully, "Since you are impervious
to argument, you shall hear with your ears how by means of my two voices
I reveal my shape to my Wives, who are at this moment six thousand miles
seventy yards two feet eight inches away, the one to the North,
the other to the South. Listen, I call to them."
He chirruped, and then complacently continued: "My wives at this
moment receiving the sound of one of my voice, closely followed
by the other, and perceiving that the latter reaches them after an interval
in which sound can traverse 6.457 inches, infer that one of my mouths
is 6.457 inches further from them than the other, and accordingly know
my shape to be 6.457 inches. But you will of course understand that
my wives do not make this calculation every time they hear my two voices.
They made it, once for all, before we were married. But they COULD
make it at any time. And in the same way I can estimate the shape
of any of my Male subjects by the sense of sound."
"But how," said I, "if a Man feigns a Woman's voice with one of his
two voices, or so disguises his Southern voice that it cannot
be recognized as the echo of the Northern? May not such deceptions
cause great inconvenience? And have you no means of checking frauds
of this kind by commanding your neighbouring subjects to feel one another?"
This of course was a very stupid question, for feeling could not have
answered the purpose; but I asked with the view of irritating the Monarch,
and I succeeded perfectly.
"What!" cried he in horror, "explain your meaning." "Feel, touch,
come into contact," I replied. "If you mean by FEELING," said the
King, "approaching so close as to leave no space between two individuals,
know, Stranger, that this offence is punishable in my dominions by death.
And the reason is obvious. The frail form of a Woman, being liable
to be shattered by such an approximation, must be preserved by the State;
but since Women cannot be distinguished by the sense of sight from Men,
the Law ordains universally that neither Man nor Woman shall be
approached so closely as to destroy the interval between the approximator
and the approximated.
"And indeed what possible purpose would be served by this illegal
and unnatural excess of approximation which you call TOUCHING,
when all the ends of so brutal and course a process are attained
at once more easily and more exactly by the sense of hearing?
As to your suggested danger of deception, it is non-existent:
for the Voice, being the essence of one's Being, cannot be
thus changed at will. But come, suppose that I had the power
of passing through solid things, so that I could penetrate my subjects,
one after another, even to the number of a billion, verifying the size
and distance of each by the sense of FEELING: How much time and energy
would be wasted in this clumsy and inaccurate method! Whereas now,
in one moment of audition, I take as it were the census and statistics,
local, corporeal, mental and spiritual, of every living being in Lineland.
Hark, only hark!"
So saying he paused and listened, as if in an ecstasy, to a sound
which seemed to me no better than a tiny chirping from an innumerable
multitude of lilliputian grasshoppers.
"Truly," replied I, "your sense of hearing serves you in good stead,
and fills up many of your deficiencies. But permit me to point out
that your life in Lineland must be deplorably dull. To see nothing
but a Point! Not even to be able to contemplate a Straight Line!
Nay, not even to know what a Straight Line is! To see, yet to be cut
off from those Linear prospects which are vouchsafed to us in Flatland!
Better surely to have no sense of sight at all than to see so little!
I grant you I have not your discriminative faculty of hearing;
for the concert of all Lineland which gives you such intense pleasure,
is to me no better than a multitudinous twittering or chirping.
But at least I can discern, by sight, a Line from a Point.
And let me prove it. Just before I came into your kingdom,
I saw you dancing from left to right, and then from right to left,
with Seven Men and a Woman in your immediate proximity on the left,
and eight Men and two Women on your right. Is not this correct?"
"It is correct," said the King, "so far as the numbers and sexes
are concerned, though I know not what you mean by `right' and `left.'
But I deny that you saw these things. For how could you see the Line,
that is to say the inside, of any Man? But you must have heard these
things, and then dreamed that you saw them. And let me ask what you
mean by those words `left' and `right.' I suppose it is your way
of saying Northward and Southward."
"Not so," replied I; "besides your motion of Northward and Southward,
there is another motion which I call from right to left."
King. Exhibit to me, if you please, this motion from left to right.
I. Nay, that I cannot do, unless you could step out of your Line altogether.
King. Out of my Line? Do you mean out of the world? Out of Space?
I. Well, yes. Out of YOUR world. Out of YOUR Space. For your
Space is not the true Space. True Space is a Plane; but your
Space is only a Line.
King. If you cannot indicate this motion from left to right by yourself
moving in it, then I beg you to describe it to me in words.
I. If you cannot tell your right side from your left, I fear
that no words of mine can make my meaning clearer to you.
But surely you cannot be ignorant of so simple a distinction.
King. I do not in the least understand you.
I. Alas! How shall I make it clear? When you move straight on,
does it not sometimes occur to you that you COULD move in some other way,
turning your eye round so as to look in the direction towards which your
side is now fronting? In other words, instead of always moving
in the direction of one of your extremities, do you never feel
a desire to move in the direction, so to speak, of your side?
King. Never. And what do you mean? How can a man's inside "front"
in any direction? Or how can a man move in the direction of his inside?
I. Well then, since words cannot explain the matter, I will try deeds,
and will move gradually out of Lineland in the direction which I desire
to indicate to you.
At the word I began to move my body out of Lineland. As long
as any part of me remained in his dominion and in his view, the King
kept exclaiming, "I see you, I see you still; you are not moving."
But when I had at last moved myself out of his Line, he cried in his
shrillest voice, "She is vanished; she is dead." "I am not dead,"
replied I; "I am simply out of Lineland, that is to say, out of the
Straight Line which you call Space, and in the true Space, where I can
see things as they are. And at this moment I can see your Line,
or side--or inside as you are pleased to call it; and I can see also
the Men and Women on the North and South of you, whom I will now enumerate,
describing their order, their size, and the interval between each."
When I had done this at great length, I cried triumphantly,
"Does that at last convince you?" And, with that, I once more
entered Lineland, taking up the same position as before.
But the Monarch replied, "If you were a Man of sense--though, as
you appear to have only one voice I have little doubt you are not a
Man but a Woman--but, if you had a particle of sense, you would
listen to reason. You ask me to believe that there is another Line
besides that which my senses indicate, and another motion besides that
of which I am daily conscious. I, in return, ask you to describe
in words or indicate by motion that other Line of which you speak.
Instead of moving, you merely exercise some magic art of vanishing
and returning to sight; and instead of any lucid description of your
new World, you simply tell me the numbers and sizes of some forty
of my retinue, facts known to any child in my capital. Can anything
be more irrational or audacious? Acknowledge your folly or depart
from my dominions."
Furious at his perversity, and especially indignant that he professed
to be ignorant of my sex, I retorted in no measured terms, "Besotted Being!
You think yourself the perfection of existence, while you are in reality
the most imperfect and imbecile. You profess to see, whereas you see
nothing but a Point! You plume yourself on inferring the existence
of a Straight Line; but I CAN SEE Straight Lines, and infer the existence
of Angles, Triangles, Squares, Pentagons, Hexagons, and even Circles.
Why waste more words? Suffice it that I am the completion of your
incomplete self. You are a Line, but I am a Line of Lines called
in my country a Square: and even I, infinitely superior though
I am to you, am of little account among the great nobles of Flatland,
whence I have come to visit you, in the hope of enlightening your ignorance."
Hearing these words the King advanced towards me with a menacing cry
as if to pierce me through the diagonal; and in that same movement
there arose from myriads of his subjects a multitudinous war-cry,
increasing in vehemence till at last methought it rivalled the roar
of an army of a hundred thousand Isosceles, and the artillery
of a thousand Pentagons. Spell-bound and motionless, I could
neither speak nor move to avert the impending destruction;
and still the noise grew louder, and the King came closer,
when I awoke to find the breakfast-bell recalling me
to the realities of Flatland.
SECTION 15 Concerning a Stranger from Spaceland
From dreams I proceed to facts.
It was the last day of our 1999th year of our era.
The patterning of the rain had long ago announced nightfall;
and I was sitting (footnote 3) in the company of my wife,
musing on the events of the past and the prospects of the coming year,
the coming century, the coming Millennium.
My four Sons and two orphan Grandchildren had retired to their
several apartments; and my wife alone remained with me to see
the old Millennium out and the new one in.
I was rapt in thought, pondering in my mind some words that had
casually issued from the mouth of my youngest Grandson, a most
promising young Hexagon of unusual brilliancy and perfect angularity.
His uncles and I had been giving him his usual practical lesson in
Sight Recognition, turning ourselves upon our centres, now rapidly,
now more slowly, and questioning him as to our positions; and his
answers had been so satisfactory that I had been induced to reward him
by giving him a few hints on Arithmetic, as applied to Geometry.
Taking nine Squares, each an inch every way, I had put them together
so as to make one large Square, with a side of three inches,
and I had hence proved to my little Grandson that--though it
was impossible for us to SEE the inside of the Square--
yet we might ascertain the number of square inches in a Square
by simply squaring the number of inches in the side: "and thus,"
said I, "we know that three-to-the-second, or nine, represents the
number of square inches in a Square whose side is three inches long."
The little Hexagon meditated on this a while and then said to me;
"But you have been teaching me to raise numbers to the third power:
I suppose three-to-the-third must mean something in Geometry; what does
it mean?" "Nothing at all," replied I, "not at least in Geometry;
for Geometry has only Two Dimensions." And then I began to shew the boy
how a Point by moving through a length of three inches makes a Line of
three inches, which may be represented by three; and how a Line of three
inches, moving parallel to itself through a length of three inches,
makes a Square of three inches every way, which may be represented
Upon this, my Grandson, again returning to his former suggestion,
took me up rather suddenly and exclaimed, "Well, then, if a Point by
moving three inches, makes a Line of three inches represented by three;
and if a straight Line of three inches, moving parallel to itself,
makes a Square of three inches every way, represented by three-to-the-second;
it must be that a Square of three inches every way, moving somehow parallel
to itself (but I don't see how) must make Something else (but I don't see what)
of three inches every way--and this must be represented by three-to-the-third."
"Go to bed," said I, a little ruffled by this interruption:
"if you would talk less nonsense, you would remember more sense."
So my Grandson had disappeared in disgrace; and there I sat by my
Wife's side, endeavouring to form a retrospect of the year 1999 and of
the possibilities of the year 2000; but not quite able to shake of the
thoughts suggested by the prattle of my bright little Hexagon. Only a
few sands now remained in the half-hour glass. Rousing myself from my
reverie I turned the glass Northward for the last time in the old
Millennium; and in the act, I exclaimed aloud, "The boy is a fool."
Straightway I became conscious of a Presence in the room, and a
chilling breath thrilled through my very being. "He is no such thing,"
cried my Wife, "and you are breaking the Commandments in thus
dishonouring your own Grandson." But I took no notice of her.
Looking around in every direction I could see nothing; yet still
I FELT a Presence, and shivered as the cold whisper came again.
I started up. "What is the matter?" said my Wife, "there is no draught;
what are you looking for? There is nothing." There was nothing;
and I resumed my seat, again exclaiming, "The boy is a fool, I say;
three- to-the-third can have no meaning in Geometry."
At once there came a distinctly audible reply,
"The boy is not a fool; and three-to-the-third
has an obvious Geometrical meaning."
My Wife as well as myself heard the words, although she did not
understand their meaning, and both of us sprang forward in the direction
of the sound. What was our horror when we saw before us a Figure!
At the first glance it appeared to be a Woman, seen sideways;
but a moment's observation shewed me that the extremities passed
into dimness too rapidly to represent one of the Female Sex;
and I should have thought it a Circle, only that it seemed
to change its size in a manner impossible for a Circle
or for any regular Figure of which I had had experience.
But my Wife had not my experience, nor the coolness necessary
to note these characteristics. With the usual hastiness
and unreasoning jealousy of her Sex, she flew at once
to the conclusion that a Woman had entered the house
through some small aperture. "How comes this person here?"
she exclaimed, "you promised me, my dear, that there should
be no ventilators in our new house." "Nor are they any," said I;
"but what makes you think that the stranger is a Woman?
I see by my power of Sight Recognition --"
"Oh, I have no patience with your Sight Recognition," replied she,
"`Feeling is believing' and `A Straight Line to the touch is worth
a Circle to the sight'"--two Proverbs, very common with the Frailer
Sex in Flatland.
"Well," said I, for I was afraid of irritating her, "if it must be so,
demand an introduction." Assuming her most gracious manner, my Wife
advanced towards the Stranger, "Permit me, Madam to feel and be felt by--"
then, suddenly recoiling, "Oh! it is not a Woman, and there are no angles
either, not a trace of one. Can it be that I have so misbehaved
to a perfect Circle?"
"I am indeed, in a certain sense a Circle," replied the Voice,
"and a more perfect Circle than any in Flatland; but to speak more
accurately, I am many Circles in one." Then he added more mildly,
"I have a message, dear Madam, to your husband, which I must not
deliver in your presence; and, if you would suffer us to retire
for a few minutes --" But my wife would not listen to the proposal
that our august Visitor should so incommode himself, and assuring
the Circle that the hour of her own retirement had long passed,
with many reiterated apologies for her recent indiscretion,
she at last retreated to her apartment.
I glanced at the half-hour glass. The last sands had fallen.
The third Millennium had begun.
Footnote 3. When I say "sitting," of course I do not mean any change
of attitude such as you in Spaceland signify by that word; for as we
have no feet, we can no more "sit" nor "stand" (in your sense of the
word) than one of your soles or flounders.
Nevertheless, we perfectly well recognize the different mental
states of volition implied by "lying," "sitting," and "standing,"
which are to some extent indicated to a beholder by a slight increase
of lustre corresponding to the increase of volition.
But on this, and a thousand other kindred subjects, time forbids me to dwell.
SECTION 16 How the Stranger vainly endeavoured to reveal to me
in words the mysteries of Spaceland
As soon as the sound of the Peace-cry of my departing Wife had died away,
I began to approach the Stranger with the intention of taking a nearer
view and of bidding him be seated: but his appearance struck me dumb
and motionless with astonishment. Without the slightest symptoms
of angularity he nevertheless varied every instant with graduations
of size and brightness scarcely possible for any Figure within the scope
of my experience. The thought flashed across me that I might have before
me a burglar or cut-throat, some monstrous Irregular Isosceles, who,
by feigning the voice of a Circle, had obtained admission somehow
into the house, and was now preparing to stab me with his acute angle.
In a sitting-room, the absence of Fog (and the season happened
to be remarkably dry), made it difficult for me to trust to Sight
Recognition, especially at the short distance at which I was standing.
Desperate with fear, I rushed forward with an unceremonious, "You must
permit me, Sir --" and felt him. My Wife was right. There was not
the trace of an angle, not the slightest roughness or inequality:
never in my life had I met with a more perfect Circle. He remained
motionless while I walked around him, beginning from his eye
and returning to it again. Circular he was throughout,
a perfectly satisfactory Circle; there could not be a doubt of it.
Then followed a dialogue, which I will endeavour to set down as near
as I can recollect it, omitting only some of my profuse apologies--
for I was covered with shame and humiliation that I, a Square,
should have been guilty of the impertinence of feeling a Circle.
It was commenced by the Stranger with some impatience at the
lengthiness of my introductory process.
Stranger. Have you felt me enough by this time? Are you not
introduced to me yet?
I. Most illustrious Sir, excuse my awkwardness, which arises not
from ignorance of the usages of polite society, but from a little
surprise and nervousness, consequent on this somewhat unexpected visit.
And I beseech you to reveal my indiscretion to no one, and especially
not to my Wife. But before your Lordship enters into further
communications, would he deign to satisfy the curiosity
of one who would gladly know whence his visitor came?
Stranger. From Space, from Space, Sir: whence else?
I. Pardon me, my Lord, but is not your Lordship already in Space,
your Lordship and his humble servant, even at this moment?
Stranger. Pooh! what do you know of Space? Define Space.
I. Space, my Lord, is height and breadth indefinitely prolonged.
Stranger. Exactly: you see you do not even know what Space is.
You think it is of Two Dimensions only; but I have come to announce
to you a Third--height, breadth, and length.
I. Your Lordship is pleased to be merry. We also speak of length
and height, or breadth and thickness, thus denoting Two Dimensions
by four names.
Stranger. But I mean not only three names, but Three Dimensions.
I. Would your Lordship indicate or explain to me
in what direction is the Third Dimension, unknown to me?
Stranger. I came from it. It is up above and down below.
I. My Lord means seemingly that it is Northward and Southward.
Stranger. I mean nothing of the kind. I mean a direction
in which you cannot look, because you have no eye in your side.
I. Pardon me, my Lord, a moment's inspection will convince your Lordship
that I have a perfectly luminary at the juncture of my two sides.
Stranger: Yes: but in order to see into Space you ought to have an eye,
not on your Perimeter, but on your side, that is, on what you would probably
call your inside; but we in Spaceland should call it your side.
I. An eye in my inside! An eye in my stomach! Your Lordship jests.
Stranger. I am in no jesting humour. I tell you that I come from Space,
or, since you will not understand what Space means, from the Land
of Three Dimensions whence I but lately looked down upon your Plane
which you call Space forsooth. From that position of advantage
I discerned all that you speak of as SOLID (by which you mean
"enclosed on four sides"), your houses, your churches,
your very chests and safes, yes even your insides and stomachs,
all lying open and exposed to my view.
I. Such assertions are easily made, my Lord.
Stranger. But not easily proved, you mean. But I mean to prove mine.
When I descended here, I saw your four Sons, the Pentagons,
each in his apartment, and your two Grandsons the Hexagons;
I saw your youngest Hexagon remain a while with you and then
retire to his room, leaving you and your Wife alone.
I saw your Isosceles servants, three in number,
in the kitchen at supper, and the little Page
in the scullery. Then I came here, and how do you think I came?
I. Through the roof, I suppose.
Strange. Not so. Your roof, as you know very well, has been
recently repaired, and has no aperture by which even a Woman
could penetrate. I tell you I come from Space. Are you not convinced
by what I have told you of your children and household?
I. Your Lordship must be aware that such facts touching
the belongings of his humble servant might be easily ascertained
by any one of the neighbourhood possessing your Lordship's
ample means of information.
Stranger. (TO HIMSELF.) What must I do? Stay; one more
argument suggests itself to me. When you see a Straight Line--
your wife, for example--how many Dimensions do you attribute to her?
I. Your Lordship would treat me as if I were one of the vulgar who,
being ignorant of Mathematics, suppose that a Woman is really a Straight Line,
and only of One Dimension. No, no, my Lord; we Squares are better advised,
and are as well aware of your Lordship that a Woman, though popularly
called a Straight Line, is, really and scientifically,
a very thin Parallelogram, possessing Two Dimensions,
like the rest of us, viz., length and breadth (or thickness).
Stranger. But the very fact that a Line is visible implies
that it possesses yet another Dimension.
I. My Lord, I have just acknowledged that a Woman is broad as well as long.
We see her length, we infer her breadth; which, though very slight,
is capable of measurement.
Stranger. You do not understand me. I mean that when you see a Woman,
you ought--besides inferring her breadth--to see her length,
and to SEE what we call her HEIGHT; although the last Dimension
is infinitesimal in your country. If a Line were mere length
without "height," it would cease to occupy Space and would become invisible.
Surely you must recognize this?
I. I must indeed confess that I do not in the least understand
your Lordship. When we in Flatland see a Line, we see length
and BRIGHTNESS. If the brightness disappears, the Line is extinguished,
and, as you say, ceases to occupy Space. But am I to suppose that
your Lordship gives the brightness the title of a Dimension,
and that what we call "bright" you call "high"?
Stranger. No, indeed. By "height" I mean a Dimension like your length:
only, with you, "height" is not so easily perceptible, being extremely small.
I. My Lord, your assertion is easily put to the test. You say
I have a Third Dimension, which you call "height." Now, Dimension
implies direction and measurement. Do but measure my "height,"
or merely indicate to me the direction in which my "height" extends,
and I will become your convert. Otherwise, your Lordship's
own understand must hold me excused.
Stranger. (TO HIMSELF.) I can do neither. How shall I convince him?
Surely a plain statement of facts followed by ocular demonstration ought
to suffice. --Now, Sir; listen to me.
You are living on a Plane. What you style Flatland is the vast level
surface of what I may call a fluid, or in, the top of which you and your
countrymen move about, without rising above or falling below it.
I am not a plane Figure, but a Solid. You call me a Circle; but in
reality I am not a Circle, but an infinite number of Circles, of size
varying from a Point to a Circle of thirteen inches in diameter,
one placed on the top of the other. When I cut through your plane as
I am now doing, I make in your plane a section which you, very rightly,
call a Circle. For even a Sphere--which is my proper name in my own
country--if he manifest himself at all to an inhabitant of Flatland--
must needs manifest himself as a Circle.
Do you not remember--for I, who see all things, discerned last
night the phantasmal vision of Lineland written upon your brain--
do you not remember, I say, how when you entered the realm of Lineland,
you were compelled to manifest yourself to the King, not as a Square,
but as a Line, because that Linear Realm had not Dimensions enough
to represent the whole of you, but only a slice or section of you?
In precisely the same way, your country of Two Dimensions is not spacious
enough to represent me, a being of Three, but can only exhibit a slice
or section of me, which is what you call a Circle.
The diminished brightness of your eye indicates incredulity.
But now prepare to receive proof positive of the truth of my assertions.
You cannot indeed see more than one of my sections, or Circles, at a time;
for you have no power to raise your eye out of the plane of Flatland;
but you can at least see that, as I rise in Space, so my sections
become smaller. See now, I will rise; and the effect upon your eye
will be that my Circle will become smaller and smaller till it dwindles
to a point and finally vanishes.
There was no "rising" that I could see; but he diminished and
finally vanished. I winked once or twice to make sure that
I was not dreaming. But it was no dream. For from the depths
of nowhere came forth a hollow voice--close to my heart it seemed--
"Am I quite gone? Are you convinced now? Well, now I will gradually
return to Flatland and you shall see my section become larger and larger."
Every reader in Spaceland will easily understand that my mysterious Guest
was speaking the language of truth and even of simplicity. But to me,
proficient though I was in Flatland Mathematics, it was by no means
a simple matter. The rough diagram given above will make it clear
to any Spaceland child that the Sphere, ascending in the three positions
indicated there, must needs have manifested himself to me, or to any
Flatlander, as a Circle, at first of full size, then small, and at last
very small indeed, approaching to a Point. But to me, although I saw
the facts before me, the causes were as dark as ever. All that I could
comprehend was, that the Circle had made himself smaller and vanished,
and that he had now re-appeared and was rapidly making himself larger.
When he regained his original size, he heaved a deep sigh; for he
perceived by my silence that I had altogether failed to comprehend
him. And indeed I was now inclining to the belief that he must be
no Circle at all, but some extremely clever juggler; or else that
the old wives' tales were true, and that after all there were such
people as Enchanters and Magicians.
After a long pause he muttered to himself, "One resource alone remains,
if I am not to resort to action. I must try the method of Analogy."
Then followed a still longer silence, after which he continued our dialogue.
Sphere. Tell me, Mr. Mathematician; if a Point moves Northward,
and leaves a luminous wake, what name would you give to the wake?
I. A straight Line.
Sphere. And a straight Line has how many extremities?
Sphere. Now conceive the Northward straight Line moving parallel
to itself, East and West, so that every point in it leaves behind
it the wake of a straight Line. What name will you give to the Figure
thereby formed? We will suppose that it moves through a distance equal
to the original straight line. --What name, I say?
I. A square.
Sphere. And how many sides has a Square? How many angles?
I. Four sides and four angles.
Sphere. Now stretch your imagination a little, and conceive a Square
in Flatland, moving parallel to itself upward.
I. What? Northward?
Sphere. No, not Northward; upward; out of Flatland altogether.
If it moved Northward, the Southern points in the Square
would have to move through the positions previously occupied
by the Northern points. But that is not my meaning.
I mean that every Point in you--for you are a Square and will serve
the purpose of my illustration--every Point in you, that is to say
in what you call your inside, is to pass upwards through Space
in such a way that no Point shall pass through the position previously
occupied by any other Point; but each Point shall describe a straight
Line of its own. This is all in accordance with Analogy;
surely it must be clear to you.
Restraining my impatience--for I was now under a strong temptation
to rush blindly at my Visitor and to precipitate him into Space,
or out of Flatland, anywhere, so that I could get rid of him--I replied:--
"And what may be the nature of the Figure which I am to shape out
by this motion which you are pleased to denote by the word `upward'?
I presume it is describable in the language of Flatland."
Sphere. Oh, certainly. It is all plain and simple, and in strict
accordance with Analogy--only, by the way, you must not speak of the
result as being a Figure, but as a Solid. But I will describe it to you.
Or rather not I, but Analogy.
We began with a single Point, which of course--being itself a Point--
has only ONE terminal Point.
One Point produces a Line with TWO terminal Points.
One Line produces a Square with FOUR terminal Points.
Now you can give yourself the answer to your own question: 1, 2,
4, are evidently in Geometrical Progression. What is the next number?
Sphere. Exactly. The one Square produces a SOMETHING-WHICH-YOU-
DO-NOT-AS-YET-KNOW-A-NAME-FOR-BUT-WHICH-WE-CALL-A-CUBE with EIGHT
terminal Points. Now are you convinced?
I. And has this Creature sides, as well as Angles or what you call
Sphere. Of course; and all according to Analogy. But, by the way,
not what YOU call sides, but what WE call sides. You would call them SOLIDS.
I. And how many solids or sides will appertain to this Being whom
I am to generate by the motion of my inside in an "upward" direction,
and whom you call a Cube?
Sphere. How can you ask? And you a mathematician! The side of anything
is always, if I may so say, one Dimension behind the thing. Consequently,
as there is no Dimension behind a Point, a Point has 0 sides; a Line,
if I may so say, has 2 sides (for the points of a Line may be called
by courtesy, its sides); a Square has 4 sides; 0, 2, 4; what Progression
do you call that?
Sphere. And what is the next number?
Sphere. Exactly. Then you see you have answered your own question.
The Cube which you will generate will be bounded by six sides,
that is to say, six of your insides. You see it all now, eh?
"Monster," I shrieked, "be thou juggler, enchanter, dream, or
devil, no more will I endure thy mockeries. Either thou or I must
perish." And saying these words I precipitated myself upon him.
SECTION 17 How the Sphere, having in vain tried words, resorted to deeds
It was in vain. I brought my hardest right angle into violent collision
with the Stranger, pressing on him with a force sufficient to have destroyed
any ordinary Circle: but I could feel him slowly and unarrestably slipping
from my contact; not edging to the right nor to the left, but moving somehow
out of the world, and vanishing into nothing. Soon there was a blank.
But still I heard the Intruder's voice.
Sphere. Why will you refuse to listen to reason? I had hoped to find
in you--as being a man of sense and an accomplished mathematician--
a fit apostle for the Gospel of the Three Dimensions, which I am allowed
to preach once only in a thousand years: but now I know not how
to convince you. Stay, I have it. Deeds, and not words,
shall proclaim the truth. Listen, my friend.
I have told you I can see from my position in Space the inside
of all things that you consider closed. For example, I see in yonder
cupboard near which you are standing, several of what you call boxes
(but like everything else in Flatland, they have no tops or bottom)
full of money; I see also two tablets of accounts. I am about
to descend into that cupboard and to bring you one of those tablets.
I saw you lock the cupboard half an hour ago, and I know you have
the key in your possession. But I descend from Space; the doors, you see,
remain unmoved. Now I am in the cupboard and am taking the tablet.
Now I have it. Now I ascend with it.
I rushed to the closet and dashed the door open. One of the tablets
was gone. With a mocking laugh, the Stranger appeared in the other
corner of the room, and at the same time the tablet appeared upon the floor.
I took it up. There could be no doubt--it was the missing tablet.
I groaned with horror, doubting whether I was not out of my sense;
but the Stranger continued: "Surely you must now see that my explanation,
and no other, suits the phenomena. What you call Solid things are really
superficial; what you call Space is really nothing but a great Plane.
I am in Space, and look down upon the insides of the things of which
you only see the outsides. You could leave the Plane yourself,
if you could but summon up the necessary volition. A slight upward
or downward motion would enable you to see all that I can see.
"The higher I mount, and the further I go from your Plane,
the more I can see, though of course I see it on a smaller scale.
For example, I am ascending; now I can see your neighbour the Hexagon
and his family in their several apartments; now I see the inside of
the Theatre, ten doors off, from which the audience is only just departing;
and on the other side a Circle in his study, sitting at his books.
Now I shall come back to you. And, as a crowning proof, what do
you say to my giving you a touch, just the least touch, in your stomach?
It will not seriously injure you, and the slight pain you may suffer
cannot be compared with the mental benefit you will receive."
Before I could utter a word of remonstrance, I felt a shooting pain
in my inside, and a demoniacal laugh seemed to issue from within me.
A moment afterwards the sharp agony had ceased, leaving nothing but
a dull ache behind, and the Stranger began to reappear, saying,
as he gradually increased in size, "There, I have not hurt you much,
have I? If you are not convinced now, I don't know what will convince you.
What say you?"
My resolution was taken. It seemed intolerable that I should endure
existence subject to the arbitrary visitations of a Magician who could
thus play tricks with one's very stomach. If only I could in any way
manage to pin him against the wall till help came!
Once more I dashed my hardest angle against him, at the same time
alarming the whole household by my cries for aid. I believe,
at the moment of my onset, the Stranger had sunk below our Plane,
and really found difficulty in rising. In any case he remained motionless,
while I, hearing, as I thought, the sound of some help approaching,
pressed against him with redoubled vigor, and continued to shout
A convulsive shudder ran through the Sphere. "This must not be,"
I thought I heard him say: "either he must listen to reason,
or I must have recourse to the last resource of civilization."
Then, addressing me in a louder tone, he hurriedly exclaimed, "Listen:
no stranger must witness what you have witnessed. Send your Wife back
at once, before she enters the apartment. The Gospel of Three Dimensions
must not be thus frustrated. Not thus must the fruits of one thousand
years of waiting be thrown away. I hear her coming. Back! back!
Away from me, or you must go with me--wither you know not--into
the Land of Three Dimensions!"
"Fool! Madman! Irregular!" I exclaimed; "never will I release thee;
thou shalt pay the penalty of thine impostures."
"Ha! Is it come to this?" thundered the Stranger: "then meet your fate:
out of your Plane you go. Once, twice, thrice! `Tis done!"
SECTION 18 How I came to Spaceland, and what I saw there
An unspeakable horror seized me. There was a darkness; then a dizzy,
sickening sensation of sight that was not like seeing; I saw a Line
that was no Line; Space that was not Space: I was myself, and not myself.
When I could find voice, I shrieked loud in agony, "Either this is madness
or it is Hell." "It is neither," calmly replied the voice of the Sphere,
"it is Knowledge; it is Three Dimensions: open your eye once again
and try to look steadily."
I looked, and, behold, a new world! There stood before me,
visibly incorporate, all that I had before inferred, conjectured,
dreamed, of perfect Circular beauty. What seemed the centre
of the Stranger's form lay open to my view: yet I could see no heart,
lungs, nor arteries, only a beautiful harmonious Something--
for which I had no words; but you, my Readers in Spaceland,
would call it the surface of the Sphere.
Prostrating myself mentally before my Guide, I cried, "How is it,
O divine ideal of consummate loveliness and wisdom that I see thy
inside, and yet cannot discern thy heart, thy lungs, thy arteries,
thy liver?" "What you think you see, you see not," he replied;
"it is not giving to you, nor to any other Being, to behold
my internal parts. I am of a different order of Beings
from those in Flatland. Were I a Circle, you could
discern my intestines, but I am a Being, composed
as I told you before, of many Circles, the Many in the One,
called in this country a Sphere. And, just as the outside
of a Cube is a Square, so the outside of a Sphere represents
the appearance of a Circle."
Bewildered though I was by my Teacher's enigmatic utterance,
I no longer chafed against it, but worshipped him in silent adoration.
He continued, with more mildness in his voice. "Distress not yourself
if you cannot at first understand the deeper mysteries of Spaceland.
By degrees they will dawn upon you. Let us begin by casting back
a glance at the region whence you came. Return with me a while to
the plains of Flatland and I will shew you that which you have often
reasoned and thought about, but never seen with the sense of sight--
a visible angle." "Impossible!" I cried; but, the Sphere leading the way,
I followed as if in a dream, till once more his voice arrested me:
"Look yonder, and behold your own Pentagonal house, and all its inmates."
I looked below, and saw with my physical eye all that domestic
individuality which I had hitherto merely inferred with
the understanding. And how poor and shadowy was the inferred conjecture
in comparison with the reality which I now behold! My four Sons
calmly asleep in the North-Western rooms, my two orphan Grandsons
to the South; the Servants, the Butler, my Daughter, all in their
several apartments. Only my affectionate Wife, alarmed by my continued
absence, had quitted her room and was roving up and down in the Hall,
anxiously awaiting my return. Also the Page, aroused by my cries,
had left his room, and under pretext of ascertaining whether I had
fallen somewhere in a faint, was prying into the cabinet in my study.
All this I could now SEE, not merely infer; and as we came nearer
and nearer, I could discern even the contents of my cabinet, and the
two chests of gold, and the tablets of which the Sphere had made mention.
Touched by my Wife's distress, I would have sprung downward
to reassure her, but I found myself incapable of motion.
"Trouble not yourself about your Wife," said my Guide:
"she will not be long left in anxiety; meantime,
let us take a survey of Flatland."
Once more I felt myself rising through space. It was even as
the Sphere had said. The further we receded from the object we beheld,
the larger became the field of vision. My native city, with
the interior of every house and every creature therein, lay open
to my view in miniature. We mounted higher, and lo, the secrets
of the earth, the depths of the mines and inmost caverns of the hills,
were bared before me.
Awestruck at the sight of the mysteries of the earth, thus unveiled
before my unworthy eye, I said to my Companion, "Behold, I am become
as a God. For the wise men in our country say that to see all things,
or as they express it, OMNIVIDENCE, is the attribute of God alone."
There was something of scorn in the voice of my Teacher as he made answer:
"it is so indeed? Then the very pick-pockets and cut-throats
of my country are to be worshipped by your wise men as being Gods:
for there is not one of them that does not see as much as you see now.
But trust me, your wise men are wrong."
I. Then is omnividence the attribute of others besides Gods?
Sphere. I do not know. But, if a pick-pocket or a cut-throat
of our country can see everything that is in your country, surely
that is no reason why the pick-pocket or cut-throat should be accepted
by you as a God. This omnividence, as you call it--it is not a common word
in Spaceland--does it make you more just, more merciful, less selfish,
more loving? Not in the least. Then how does it make you more divine?
I. "More merciful, more loving!" But these are the qualities of women!
And we know that a Circle is a higher Being than a Straight Line,
in so far as knowledge and wisdom are more to be esteemed than mere affection.
Sphere. It is not for me to classify human faculties according to merit.
Yet many of the best and wisest in Spaceland think more of the affections
than of the understand, more of your despised Straight Lines than of your
belauded Circles. But enough of this. Look yonder. Do you know
I looked, and afar off I saw an immense Polygonal structure,
in which I recognized the General Assembly Hall of the States
of Flatland, surrounded by dense lines of Pentagonal buildings
at right angles to each other, which I knew to be streets;
and I perceived that I was approaching the great Metropolis.
"Here we descend," said my Guide. It was now morning, the first
hour of the first day of the two thousandth year of our era.
Acting, as was their wont, in strict accordance with precedent,
the highest Circles of the realm were meeting in solemn conclave,
as they had met on the first hour of the first day of the year 1000,
and also on the first hour of the first day of the year 0.
The minutes of the previous meetings were now read by one whom
I at once recognized as my brother, a perfectly Symmetrical Square,
and the Chief Clerk of the High Council. It was found recorded on
each occasion that: "Whereas the States had been troubled by divers
ill-intentioned persons pretending to have received revelations
from another World, and professing to produce demonstrations whereby
they had instigated to frenzy both themselves and others, it had been
for this cause unanimously resolved by the Grand Council that on the
first day of each millenary, special injunctions be sent to the Prefects
in the several districts of Flatland, to make strict search for such
misguided persons, and without formality of mathematical examination,
to destroy all such as were Isosceles of any degree, to scourge
and imprison any regular Triangle, to cause any Square or Pentagon
to be sent to the district Asylum, and to arrest any one of higher rank,
sending him straightway to the Capital to be examined and judged
by the Council."
"You hear your fate," said the Sphere to me, while the Council
was passing for the third time the formal resolution. "Death or
imprisonment awaits the Apostle of the Gospel of Three Dimensions."
"Not so," replied I, "the matter is now so clear to me, the nature of real
space so palpable, that methinks I could make a child understand it.
Permit me but to descend at this moment and enlighten them."
"Not yet," said my Guide, "the time will come for that.
Meantime I must perform my mission. Stay thou there in thy place."
Saying these words, he leaped with great dexterity into the sea
(if I may so call it) of Flatland, right in the midst of the ring
of Counsellors. "I come," said he, "to proclaim that there is a land
of Three Dimensions."
I could see many of the younger Counsellors start back in manifest horror,
as the Sphere's circular section widened before them. But on a sign from
the presiding Circle--who shewed not the slightest alarm or surprise--
six Isosceles of a low type from six different quarters rushed upon the Sphere.
"We have him," they cried; "No; yes; we have him still! he's going! he's gone!"
"My Lords," said the President to the Junior Circles of the Council,
"there is not the slightest need for surprise; the secret archives,
to which I alone have access, tell me that a similar occurrence
happened on the last two millennial commencements. You will,
of course, say nothing of these trifles outside the Cabinet."
Raising his voice, he now summoned the guards. "Arrest the policemen;
gag them. You know your duty." After he had consigned to their fate
the wretched policemen--ill-fated and unwilling witnesses
of a State-secret which they were not to be permitted to reveal--
he again addressed the Counsellors. "My Lords, the business of the
Council being concluded, I have only to wish you a happy New Year."
Before departing, he expressed, at some length, to the Clerk,
my excellent but most unfortunate brother, his sincere regret that,
in accordance with precedent and for the sake of secrecy, he must condemn
him to perpetual imprisonment, but added his satisfaction that,
unless some mention were made by him of that day's incident,
his life would be spared.
SECTION 19 How, though the Sphere shewed me other mysteries of
Spaceland, I still desire more; and what came of it
When I saw my poor brother led away to imprisonment, I attempted to leap
down into the Council Chamber, desiring to intercede on his behalf,
or at least bid him farewell. But I found that I had no motion of my own.
I absolutely depended on the volition of my Guide, who said in gloomy tones,
"Heed not thy brother; haply thou shalt have ample time hereafter
to condole with him. Follow me."
Once more we ascended into space. "Hitherto," said the Sphere,
"I have shewn you naught save Plane Figures and their interiors.
Now I must introduce you to Solids, and reveal to you the plan upon which
they are constructed. Behold this multitude of moveable square cards.
See, I put one on another, not, as you supposed, Northward of the other,
but ON the other. Now a second, now a third. See, I am building up
a Solid by a multitude of Squares parallel to one another.
Now the Solid is complete, being as high as it is long and broad,
and we call it a Cube."
"Pardon me, my Lord," replied I; "but to my eye the appearance
is as of an Irregular Figure whose inside is laid open to view;
in other words, methinks I see no Solid, but a Plane such as we
infer in Flatland; only of an Irregularity which betokens some
monstrous criminal, so that the very sight of it is painful to my eyes."
"True," said the Sphere; "it appears to you a Plane, because you
are not accustomed to light and shade and perspective; just as in
Flatland a Hexagon would appear a Straight Line to one who has not
the Art of Sight Recognition. But in reality it is a Solid,
as you shall learn by the sense of Feeling."
He then introduced me to the Cube, and I found that this marvellous
Being was indeed no Plane, but a Solid; and that he was endowed with
six plane sides and eight terminal points called solid angles;
and I remembered the saying of the Sphere that just such a Creature
as this would be formed by the Square moving, in Space, parallel to himself:
and I rejoiced to think that so insignificant a Creature as I could
in some sense be called the Progenitor of so illustrious an offspring.
But still I could not fully understand the meaning of what my Teacher
had told me concerning "light" and "shade" and "perspective";
and I did not hesitate to put my difficulties before him.
Were I to give the Sphere's explanation of these matters, succinct
and clear though it was, it would be tedious to an inhabitant of Space,
who knows these things already. Suffice it, that by his lucid statements,
and by changing the position of objects and lights, and by allowing me
to feel the several objects and even his own sacred Person, he at last
made all things clear to me, so that I could now readily distinguish
between a Circle and a Sphere, a Plane Figure and a Solid.
This was the Climax, the Paradise, of my strange eventful History.
Henceforth I have to relate the story of my miserable Fall:--most miserable,
yet surely most undeserved! For why should the thirst for knowledge
be aroused, only to be disappointed and punished? My volition shrinks
from the painful task of recalling my humiliation; yet, like a second
Prometheus, I will endure this and worse, if by any means I may arouse
in the interiors of Plane and Solid Humanity a spirit of rebellion against
the Conceit which would limit our Dimensions to Two or Three or any number
short of Infinity. Away then with all personal considerations!
Let me continue to the end, as I began, without further digressions
or anticipations, pursuing the plain path of dispassionate History.
The exact facts, the exact words,--and they are burnt in upon my brain,
--shall be set down without alteration of an iota; and let my Readers
judge between me and Destiny.
The Sphere would willingly have continued his lessons by
indoctrinating me in the conformation of all regular Solids,
Cylinders, Cones, Pyramids, Pentahedrons, Hexahedrons, Dodecahedrons,
and Spheres: but I ventured to interrupt him. Not that I was wearied
of knowledge. On the contrary, I thirsted for yet deeper and fuller
draughts than he was offering to me.
"Pardon me," said I, "O Thou Whom I must no longer address as the Perfection
of all Beauty; but let me beg thee to vouchsafe thy servant a sight
of thine interior."
Sphere. My what?
I. Thine interior: thy stomach, thy intestines.
Sphere. Whence this ill-timed impertinent request? And what mean
you by saying that I am no longer the Perfection of all Beauty?
I. My Lord, your own wisdom has taught me to aspire to One even more great,
more beautiful, and more closely approximate to Perfection than yourself.
As you yourself, superior to all Flatland forms, combine many Circles in One,
so doubtless there is One above you who combines many Spheres
in One Supreme Existence, surpassing even the Solids of Spaceland.
And even as we, who are now in Space, look down on Flatland
and see the insides of all things, so of a certainty there is yet
above us some higher, purer region, whither thou dost surely purpose
to lead me--O Thou Whom I shall always call, everywhere and in all Dimensions,
my Priest, Philosopher, and Friend--some yet more spacious Space,
some more dimensionable Dimensionality, from the vantage-ground
of which we shall look down together upon the revealed insides
of Solid things, and where thine own intestines, and those
of thy kindred Spheres, will lie exposed to the view of the poor
wandering exile from Flatland, to whom so much has already been vouchsafed.
Sphere. Pooh! Stuff! Enough of this trifling! The time is short,
and much remains to be done before you are fit to proclaim the Gospel
of Three Dimensions to your blind benighted countrymen in Flatland.
I. Nay, gracious Teacher, deny me not what I know it is in thy
power to reform. Grant me but one glimpse of thine interior,
and I am satisfied for ever, remaining henceforth thy docile pupil,
thy unemacipable slave, ready to receive all thy teachings and to feed
upon the words that fall from thy lips.
Sphere. Well, then, to content and silence you, let me say at once,
I would shew you what you wish if I could; but I cannot. Would you
have me turn my stomach inside out to oblige you?
I. But my Lord has shewn me the intestines of all my countrymen in
the Land of Two Dimensions by taking me with him into the Land of Three.
What therefore more easy than now to take his servant on a second journey
into the blessed region of the Fourth Dimension, where I shall look down
with him once more upon this land of Three Dimensions, and see the inside
of every three-dimensioned house, the secrets of the solid earth,
the treasures of the mines of Spaceland, and the intestines of every
solid living creature, even the noble and adorable Spheres.
Sphere. But where is this land of Four Dimensions?
I. I know not: but doubtless my Teacher knows.
Sphere. Not I. There is no such land. The very idea of it is
I. Not inconceivable, my Lord, to me, and therefore still less
inconceivable to my Master. Nay, I despair not that, even here,
in this region of Three Dimensions, your Lordship's art may make
the Fourth Dimension visible to me; just as in the Land of
Two Dimensions my Teacher's skill would fain have opened the eyes
of his blind servant to the invisible presence of a Third Dimension,
though I saw it not.
Let me recall the past. Was I not taught below that when I saw a Line
and inferred a Plane, I in reality saw a Third unrecognized Dimension,
not the same as brightness, called "height"? And does it not
now follow that, in this region, when I see a Plane and infer a Solid,
I really see a Fourth unrecognized Dimension, not the same as colour,
but existent, though infinitesimal and incapable of measurement?
And besides this, there is the Argument from Analogy of Figures.
Sphere. Analogy! Nonsense: what analogy?
I. Your Lordship tempts his servant to see whether he remembers
the revelations imparted to him. Trifle not with me, my Lord;
I crave, I thirst, for more knowledge. Doubtless we cannot SEE that
other higher Spaceland now, because we have no eye in our stomachs.
But, just as there WAS the realm of Flatland, though that poor puny
Lineland Monarch could neither turn to left nor right to discern it,
and just as there WAS close at hand, and touching my frame, the land
of Three Dimensions, though I, blind senseless wretch, had no power
to touch it, no eye in my interior to discern it, so of a surety there
is a Fourth Dimension, which my Lord perceives with the inner eye
of thought. And that it must exist my Lord himself has taught me.
Or can he have forgotten what he himself imparted to his servant?
In One Dimension, did not a moving Point produce a Line with TWO
In Two Dimensions, did not a moving Line produce a Square with FOUR
In Three Dimensions, did not a moving Square produce--did not
this eye of mine behold it--that blessed Being, a Cube, with EIGHT
And in Four Dimensions shall not a moving Cube--alas, for Analogy,
and alas for the Progress of Truth, if it be not so--shall not,
I say, the motion of a divine Cube result in a still more divine
Organization with SIXTEEN terminal points?
Behold the infallible confirmation of the Series, 2, 4, 8, 16: is
not this a Geometrical Progression? Is not this--if I might quote
my Lord's own words--"strictly according to Analogy"?
Again, was I not taught by my Lord that as in a Line there are TWO
bounding Points, and in a Square there are FOUR bounding Lines,
so in a Cube there must be SIX bounding Squares? Behold once more
the confirming Series, 2, 4, 6: is not this an Arithmetical Progression?
And consequently does it not of necessity follow that the more divine
offspring of the divine Cube in the Land of Four Dimensions,
must have 8 bounding Cubes: and is not this also, as my Lord
has taught me to believe, "strictly according to Analogy"?
O, my Lord, my Lord, behold, I cast myself in faith upon conjecture,
not knowing the facts; and I appeal to your Lordship to confirm
or deny my logical anticipations. If I am wrong, I yield,
and will no longer demand a Fourth Dimension; but,
if I am right, my Lord will listen to reason.
I ask therefore, is it, or is it not, the fact, that ere now your
countrymen also have witnessed the descent of Beings of a higher order
than their own, entering closed rooms, even as your Lordship entered mine,
without the opening of doors or windows, and appearing and vanishing at will?
On the reply to this question I am ready to stake everything. Deny it,
and I am henceforth silent. Only vouchsafe an answer.
Sphere (AFTER A PAUSE). It is reported so. But men are divided
in opinion as to the facts. And even granting the facts, they explain
them in different ways. And in any case, however great may be the
number of different explanations, no one has adopted or suggested
the theory of a Fourth Dimension. Therefore, pray have done with
this trifling, and let us return to business.
I. I was certain of it. I was certain that my anticipations
would be fulfilled. And now have patience with me and answer me
yet one more question, best of Teachers! Those who have thus appeared--
no one knows whence--and have returned--no one knows whither--
have they also contracted their sections and vanished somehow into
that more Spacious Space, whither I now entreat you to conduct me?
Sphere (MOODILY). They have vanished, certainly--if they ever appeared.
But most people say that these visions arose from the thought--you will not
understand me--from the brain; from the perturbed angularity of the Seer.
I. Say they so? Oh, believe them not. Or if it indeed be so,
that this other Space is really Thoughtland, then take me to that
blessed Region where I in Thought shall see the insides of all solid
things. There, before my ravished eye, a Cube moving in some
altogether new direction, but strictly according to Analogy, so as to
make every particle of his interior pass through a new kind of Space,
with a wake of its own--shall create a still more perfect perfection
than himself, with sixteen terminal Extra-solid angles, and Eight
solid Cubes for his Perimeter. And once there, shall we stay our
upward course? In that blessed region of Four Dimensions, shall we
linger at the threshold of the Fifth, and not enter therein? Ah, no!
Let us rather resolve that our ambition shall soar with our corporal
ascent. Then, yielding to our intellectual onset, the gates of the
Six Dimension shall fly open; after that a Seventh, and then an Eighth--
How long I should have continued I know not. In vain did the Sphere,
in his voice of thunder, reiterate his command of silence,
and threaten me with the direst penalties if I persisted.
Nothing could stem the flood of my ecstatic aspirations.
Perhaps I was to blame; but indeed I was intoxicated with
the recent draughts of Truth to which he himself had introduced me.
However, the end was not long in coming. My words were cut short
by a crash outside, and a simultaneous crash inside me,
which impelled me through space with a velocity that precluded speech.
Down! down! down! I was rapidly descending; and I knew that return
to Flatland was my doom. One glimpse, one last and never-to-be-forgotten
glimpse I had of that dull level wilderness--which was now to become
my Universe again-- spread out before my eye. Then a darkness.
Then a final, all-consummating thunder-peal; and, when I came to myself,
I was once more a common creeping Square, in my Study at home,
listening to the Peace- Cry of my approaching Wife.
SECTION 20 How the Sphere encouraged me in a Vision.
Although I had less than a minute for reflection, I felt, by a kind
of instinct, that I must conceal my experiences from my Wife.
Not that I apprehended, at the moment, any danger from her divulging
my secret, but I knew that to any Woman in Flatland the narrative of my
adventures must needs be unintelligible. So I endeavoured to reassure
her by some story, invented for the occasion, that I had accidentally
fallen through the trap-door of the cellar, and had there lain stunned.
The Southward attraction in our country is so slight that even
to a Woman my tale necessarily appeared extraordinary and well-nigh
incredible; but my Wife, whose good sense far exceeds that of the
average of her Sex, and who perceived that I was unusually excited,
did not argue with me on the subject, but insisted that I was ill
and required repose. I was glad of an excuse for retiring to my chamber
to think quietly over what had happened. When I was at last by myself,
a drowsy sensation fell on me; but before my eyes closed I endeavoured
to reproduce the Third Dimension, and especially the process by which
a Cube is constructed through the motion of a Square. It was not
so clear as I could have wished; but I remembered that it must be
"Upward, and yet not Northward," and I determined steadfastly
to retain these words as the clue which, if firmly grasped,
could not fail to guide me to the solution. So mechanically
repeating, like a charm, the words, "Upward, yet not Northward,"
I fell into a sound refreshing sleep.
During my slumber I had a dream. I thought I was once more
by the side of the Sphere, whose lustrous hue betokened that
he had exchanged his wrath against me for perfectly placability.
We were moving together towards a bright but infinitesimally small Point,
to which my Master directed my attention. As we approached, methought
there issued from it a slight humming noise as from one of your Spaceland
bluebottles, only less resonant by far, so slight indeed that even
in the perfect stillness of the Vacuum through which we soared,
the sound reached not our ears till we checked our flight
at a distance from it of something under twenty human diagonals.
"Look yonder," said my Guide, "in Flatland thou hast lived;
of Lineland thou hast received a vision; thou hast soared with me
to the heights of Spaceland; now, in order to complete the range
of thy experience, I conduct thee downward to the lowest depth of existence,
even to the realm of Pointland, the Abyss of No dimensions.
"Behold yon miserable creature. That Point is a Being like ourselves,
but confined to the non-dimensional Gulf. He is himself his own World,
his own Universe; of any other than himself he can form no conception;
he knows not Length, nor Breadth, nor Height, for he has had no
experience of them; he has no cognizance even of the number Two;
nor has he a thought of Plurality; for he is himself his One and All,
being really Nothing. Yet mark his perfect self-contentment,
and hence learn his lesson, that to be self-contented is to be vile
and ignorant, and that to aspire is better than to be blindly
and impotently happy. Now listen."
He ceased; and there arose from the little buzzing creature a tiny,
low, monotonous, but distinct tinkling, as from one of your Spaceland
phonographs, from which I caught these words, "Infinite beatitude
of existence! It is; and there is nothing else beside It."
"What," said I, "does the puny creature mean by `it'?" "He means
himself," said the Sphere: "have you not noticed before now,
that babies and babyish people who cannot distinguish themselves
from the world, speak of themselves in the Third Person? But hush!"
"It fills all Space," continued the little soliloquizing Creature,
"and what It fills, It is. What It thinks, that It utters;
and what It utters, that It hears; and It itself is Thinker, Utterer,
Hearer, Thought, Word, Audition; it is the One, and yet the All in All.
Ah, the happiness, ah, the happiness of Being!"
"Can you not startle the little thing out of its complacency?" said I.
"Tell it what it really is, as you told me; reveal to it the narrow
limitations of Pointland, and lead it up to something higher."
"That is no easy task," said my Master; "try you."
Hereon, raising by voice to the uttermost, I addressed the Point as follows:
"Silence, silence, contemptible Creature. You call yourself the
All in All, but you are the Nothing: your so-called Universe is a
mere speck in a Line, and a Line is a mere shadow as compared with--"
"Hush, hush, you have said enough," interrupted the Sphere, "now listen,
and mark the effect of your harangue on the King of Pointland."
The lustre of the Monarch, who beamed more brightly than ever upon
hearing my words, shewed clearly that he retained his complacency;
and I had hardly ceased when he took up his strain again. "Ah,
the joy, ah, the joy of Thought! What can It not achieve by thinking!
Its own Thought coming to Itself, suggestive of its disparagement,
thereby to enhance Its happiness! Sweet rebellion stirred up to result
in triumph! Ah, the divine creative power of the All in One!
Ah, the joy, the joy of Being!"
"You see," said my Teacher, "how little your words have done.
So far as the Monarch understand them at all, he accepts them as his own--
for he cannot conceive of any other except himself--and plumes himself
upon the variety of `Its Thought' as an instance of creative Power.
Let us leave this God of Pointland to the ignorant fruition of his
omnipresence and omniscience: nothing that you or I can do can rescue
him from his self-satisfaction."
After this, as we floated gently back to Flatland, I could hear
the mild voice of my Companion pointing the moral of my vision,
and stimulating me to aspire, and to teach others to aspire.
He had been angered at first--he confessed--by my ambition to soar
to Dimensions above the Third; but, since then, he had received fresh
insight, and he was not too proud to acknowledge his error to a Pupil.
Then he proceeded to initiate me into mysteries yet higher than those
I had witnessed, shewing me how to construct Extra-Solids by the motion
of Solids, and Double Extra-Solids by the motion of Extra-Solids,
and all "strictly according to Analogy," all by methods so simple,
so easy, as to be patent even to the Female Sex.
SECTION 21 How I tried to teach the Theory of Three Dimensions
to my Grandson, and with what success
I awoke rejoicing, and began to reflect on the glorious career before me.
I would go forth, methought, at once, and evangelize the whole of Flatland.
Even to Women and Soldiers should the Gospel of Three Dimensions be proclaimed.
I would begin with my Wife.
Just as I had decided on the plan of my operations, I heard the sound
of many voices in the street commanding silence. Then followed a louder voice.
It was a herald's proclamation. Listening attentively, I recognized the words
of the Resolution of the Council, enjoining the arrest, imprisonment,
or execution of any one who should pervert the minds of people by delusions,
and by professing to have received revelations from another World.
I reflected. This danger was not to be trifled with. It would
be better to avoid it by omitting all mention of my Revelation,
and by proceeding on the path of Demonstration--which after all,
seemed so simple and so conclusive that nothing would be lost by
discarding the former means. "Upward, not Northward"--was the clue to
the whole proof. It had seemed to me fairly clear before I fell asleep;
and when I first awoke, fresh from my dream, it had appeared as patent
as Arithmetic; but somehow it did not seem to me quite so obvious now.
Though my Wife entered the room opportunely at just that moment,
I decided, after we had exchanged a few words of commonplace conversation,
not to begin with her.
My Pentagonal Sons were men of character and standing, and physicians
of no mean reputation, but not great in mathematics, and, in that respect,
unfit for my purpose. But it occurred to me that a young and docile Hexagon,
with a mathematical turn, would be a most suitable pupil. Why therefore
not make my first experiment with my little precocious Grandson,
whose casual remarks on the meaning of three-to-the-third had met
with the approval of the Sphere? Discussing the matter with him,
a mere boy, I should be in perfect safety; for he would know nothing
of the Proclamation of the Council; whereas I could not feel sure
that my Sons--so greatly did their patriotism and reverence
for the Circles predominate over mere blind affection--
might not feel compelled to hand me over to the Prefect,
if they found me seriously maintaining the seditious
heresy of the Third Dimension.
But the first thing to be done was to satisfy in some way the curiosity
of my Wife, who naturally wished to know something of the reasons
for which the Circle had desired that mysterious interview,
and of the means by which he had entered the house. Without entering
into the details of the elaborate account I gave her,--an account,
I fear, not quite so consistent with truth as my Readers in Spaceland
might desire,--I must be content with saying that I succeeded
at last in persuading her to return quietly to her household duties
without eliciting from me any reference to the World of Three Dimensions.
This done, I immediately sent for my Grandson; for, to confess the truth,
I felt that all that I had seen and heard was in some strange way slipping
away from me, like the image of a half-grasped, tantalizing dream,
and I longed to essay my skill in making a first disciple.
When my Grandson entered the room I carefully secured the door.
Then, sitting down by his side and taking our mathematical tablets,--
or, as you would call them, Lines--I told him we would resume
the lesson of yesterday. I taught him once more how a Point by motion
in One Dimension produces a Line, and how a straight Line in Two
Dimensions produces a Square. After this, forcing a laugh, I said,
"And now, you scamp, you wanted to make believe that a Square may in
the same way by motion `Upward, not Northward' produce another figure,
a sort of extra square in Three Dimensions. Say that again, you young rascal."
At this moment we heard once more the herald's "O yes! O yes!"
outside in the street proclaiming the REsolution of the Council.
Young though he was, my Grandson--who was unusually intelligent
for his age, and bred up in perfect reverence for the authority
of the Circles--took in the situation with an acuteness for which
I was quite unprepared. He remained silent till the last words
of the Proclamation had died away, and then, bursting into tears,
"Dear Grandpapa," he said, "that was only my fun, and of course I meant
nothing at all by it; and we did not know anything then about the new Law;
and I don't think I said anything about the Third Dimension; and I am sure
I did not say one word about `Upward, not Northward,' for that would be
such nonsense, you know. How could a thing move Upward, and not Northward?
Upward and not Northward! Even if I were a baby, I could not be so absurd
as that. How silly it is! Ha! ha! ha!"
"Not at all silly," said I, losing my temper; "here for example,
I take this Square," and, at the word, I grasped a moveable Square,
which was lying at hand--"and I move it, you see, not Northward but
--yes, I move it Upward--that is to say, Northward but I move it
somewhere--not exactly like this, but somehow --" Here I brought
my sentence to an inane conclusion, shaking the Square about in
a purposeless manner, much to the amusement of my Grandson, who burst
out laughing louder than ever, and declared that I was not teaching
him, but joking with him; and so saying he unlocked the door and ran
out of the room. Thus ended my first attempt to convert a pupil to
the Gospel of Three Dimensions.
SECTION 22 How I then tried to diffuse the Theory of Three
Dimensions by other means, and of the result
My failure with my Grandson did not encourage me to communicate
my secret to others of my household; yet neither was I led by it
to despair of success. Only I saw that I must not wholly rely on
the catch-phrase, "Upward, not Northward," but must rather endeavour
to seek a demonstration by setting before the public a clear view
of the whole subject; and for this purpose it seemed necessary
to resort to writing.
So I devoted several months in privacy to the composition
of a treatise on the mysteries of Three Dimensions. Only,
with the view of evading the Law, if possible, I spoke not
of a physical Dimension, but of a Thoughtland whence, in theory,
a Figure could look down upon Flatland and see simultaneously
the insides of all things, and where it was possible that
there might be supposed to exist a Figure environed,
as it were, with six Squares, and containing eight terminal Points.
But in writing this book I found myself sadly hampered by
the impossibility of drawing such diagrams as were necessary
for my purpose: for of course, in our country of Flatland,
there are no tablets but Lines, and no diagrams but Lines,
all in one straight Line and only distinguishable by difference
of size and brightness; so that, when I had finished my treatise
(which I entitled, "Through Flatland to Thoughtland")
I could not feel certain that many would understand my meaning.
Meanwhile my wife was under a cloud. All pleasures palled upon me;
all sights tantalized and tempted me to outspoken treason, because
I could not compare what I saw in Two Dimensions with what it really
was if seen in Three, and could hardly refrain from making my comparisons
aloud. I neglected my clients and my own business to give myself
to the contemplation of the mysteries which I had once beheld,
yet which I could impart to no one, and found daily more difficult
to reproduce even before my own mental vision.
One day, about eleven months after my return from Spaceland, I tried
to see a Cube with my eye closed, but failed; and though I succeeded
afterwards, I was not then quite certain (nor have I been ever afterwards)
that I had exactly realized the original. This made me more melancholy
than before, and determined me to take some step; yet what, I knew not.
I felt that I would have been willing to sacrifice my life for the Cause,
if thereby I could have produced conviction. But if I could not convince
my Grandson, how could I convince the highest and most developed Circles
in the land?
And yet at times my spirit was too strong for me, and I gave vent
to dangerous utterances. Already I was considered heterodox if not
treasonable, and I was keenly alive to the danger of my position;
nevertheless I could not at times refrain from bursting out into
suspicious or half-seditious utterances, even among the highest
Polygonal or Circular society. When, for example, the question arose
about the treatment of those lunatics who said that they had received
the power of seeing the insides of things, I would quote the saying
of an ancient Circle, who declared that prophets and inspired people
are always considered by the majority to be mad; and I could not help
occasionally dropping such expressions as "the eye that discerns
the interiors of things," and "the all-seeing land"; once or twice
I even let fall the forbidden terms "the Third and Fourth Dimensions."
At last, to complete a series of minor indiscretions, at a meeting of
our Local Speculative Society held at the palace of the Prefect himself,
--some extremely silly person having read an elaborate paper exhibiting
the precise reasons why Providence has limited the number of Dimensions
to Two, and why the attribute of omnividence is assigned to the Supreme
alone--I so far forgot myself as to give an exact account of the whole
of my voyage with the Sphere into Space, and to the Assembly Hall
in our Metropolis, and then to Space again, and of my return home,
and of everything that I had seen and heard in fact or vision.
At first, indeed, I pretended that I was describing the imaginary
experiences of a fictitious person; but my enthusiasm soon forced me
to throw off all disguise, and finally, in a fervent peroration,
I exhorted all my hearers to divest themselves of prejudice
and to become believers in the Third Dimension.
Need I say that I was at once arrested and taken before the Council?
Next morning, standing in the very place where but a very few months ago
the Sphere had stood in my company, I was allowed to begin and to continue
my narration unquestioned and uninterrupted. But from the first I foresaw
my fate; for the President, noting that a guard of the better sort
of Policemen was in attendance, of angularity little, if at all,
under 55 degrees, ordered them to be relieved before I began my defence,
by an inferior class of 2 or 3 degrees. I knew only too well what that meant.
I was to be executed or imprisoned, and my story was to be kept secret
from the world by the simultaneous destruction of the officials
who had heard it; and, this being the case, the President desired
to substitute the cheaper for the more expensive victims.
After I had concluded my defence, the President, perhaps perceiving
that some of the junior Circles had been moved by evident earnestness,
asked me two questions:--
1. Whether I could indicate the direction which I meant when I used
the words "Upward, not Northward"?
2. Whether I could by any diagrams or descriptions (other than
the enumeration of imaginary sides and angles) indicate the Figure
I was pleased to call a Cube?
I declared that I could say nothing more, and that I must commit
myself to the Truth, whose cause would surely prevail in the end.
The President replied that he quite concurred in my sentiment,
and that I could not do better. I must be sentenced to perpetual
imprisonment; but if the Truth intended that I should emerge from
prison and evangelize the world, the Truth might be trusted to bring
that result to pass. Meanwhile I should be subjected to no discomfort
that was not necessary to preclude escape, and, unless I forfeited the
privilege by misconduct, I should be occasionally permitted to see my
brother who had preceded me to my prison.
Seven years have elapsed and I am still a prisoner, and--if I
except the occasional visits of my brother--debarred from all
companionship save that of my jailers. My brother is one of the best
of Squares, just sensible, cheerful, and not without fraternal
affection; yet I confess that my weekly interviews, at least
in one respect, cause me the bitterest pain. He was present when
the Sphere manifested himself in the Council Chamber; he saw the Sphere's
changing sections; he heard the explanation of the phenomena then give
to the Circles. Since that time, scarcely a week has passed during
seven whole years, without his hearing from me a repetition of the part
I played in that manifestation, together with ample descriptions of all
the phenomena in Spaceland, and the arguments for the existence of Solid
things derivable from Analogy. Yet--I take shame to be forced to confess it--
my brother has not yet grasped the nature of Three Dimensions, and frankly
avows his disbelief in the existence of a Sphere.
Hence I am absolutely destitute of converts, and, for aught that
I can see, the millennial Revelation has been made to me for nothing.
Prometheus up in Spaceland was bound for bringing down fire
for mortals, but I--poor Flatland Prometheus--lie here in prison
for bringing down nothing to my countrymen. Yet I existing the hope
that these memoirs, in some manner, I know not how, may find their way
to the minds of humanity in Some Dimension, and may stir up a race
of rebels who shall refuse to be confined to limited Dimensionality.
That is the hope of my brighter moments. Alas, it is not always so.
Heavily weights on me at times the burdensome reflection that I cannot
honestly say I am confident as to the exact shape of the once-seen,
oft-regretted Cube; and in my nightly visions the mysterious precept,
"Upward, not Northward," haunts me like a soul-devouring Sphinx.
It is part of the martyrdom which I endure for the cause of Truth
that there are seasons of mental weakness, when Cubes and Spheres
flit away into the background of scarce-possible existences;
when the Land of Three Dimensions seems almost as visionary
as the Land of One or None; nay, when even this hard wall that bars
me from my freedom, these very tablets on which I am writing,
and all the substantial realities of Flatland itself,
appear no better than the offspring of a diseased imagination,
or the baseless fabric of a dream.
PREFACE TO THE
SECOND AND REVISED
BY THE EDITOR
If my poor Flatland friend retained the vigour of mind which he enjoyed
when he began to compose these Memoirs, I should not now need to represent
him in this preface, in which he desires, fully, to return his thanks
to his readers and critics in Spaceland, whose appreciation has,
with unexpected celerity, required a second edition of this work;
secondly, to apologize for certain errors and misprints (for which,
however, he is not entirely responsible); and, thirdly, to explain
on or two misconceptions. But he is not the Square he once was.
Years of imprisonment, and the still heavier burden of general
incredulity and mockery, have combined with the thoughts and notions,
and much also of the terminology, which he acquired during his
short stay in spaceland. He has, therefore, requested me to reply
in his behalf to two special objections, one of an intellectual,
the other of a moral nature.
The first objection is, that a Flatlander, seeing a Line,
sees something that must be THICK to the eye as well as LONG
to the eye (otherwise it would not be visible, if it had not
some thickness); and consequently he ought (it is argued)
to acknowledge that his countrymen are not only long and broad,
but also (though doubtless to a very slight degree) THICK or HIGH.
This objection is plausible, and, to Spacelanders, almost irresistible,
so that, I confess, when I first heard it, I knew not what to reply.
But my poor old friend's answer appears to me completely to meet it.
"I admit," said he--when I mentioned to him this objection--
"I admit the truth of your critic's facts, but I deny his conclusions.
It is true that we have really in Flatland a Third unrecognized Dimension
called `height,' just as it also is true that you have really in Spaceland
a Fourth unrecognized Dimension, called by no name at present, but which
I will call `extra-height.' But we can no more take cognizance of our
`height' than you can of your `extra-height.' Even I--who have been in
Spaceland, and have had the privilege of understanding for twenty-four hours
the meaning of `height'--even I cannot now comprehend it, nor realize it
by the sense of sight or by any process of reason; I can but apprehend
it by faith.
"The reason is obvious. Dimension implies direction, implies
measurement, implies the more and the less. Now, all our lines
are EQUALLY and INFINITESIMALLY thick (or high, whichever you like);
consequently, there is nothing in them to lead our minds to the
conception of that Dimension. No `delicate micrometer'--as has been
suggested by one too hasty Spaceland critic--would in the least
avail us; for we should not know WHAT TO MEASURE, NOR IN WHAT DIRECTION.
When we see a Line, we see something that is long and BRIGHT;
BRIGHTNESS, as well as length, is necessary to the existence of a Line;
if the brightness vanishes, the Line is extinguished. Hence, all my
Flatland friends--when I talk to them about the unrecognized Dimension
which is somehow visible in a Line--say, `Ah, you mean BRIGHTNESS':
and when I reply, `No, I mean a real Dimension,' they at once retort,
`Then measure it, or tell us in what direction it extends'; and this
silences me, for I can do neither. Only yesterday, when the Chief Circle
(in other words our High Priest) came to inspect the State Prison
and paid me his seventh annual visit, and when for the seventh time
he put me the question, `Was I any better?' I tried to prove to him
that he was `high,' as well as long and broad, although he did not know it.
But what was his reply? `You say I am "high"; measure my "high-ness"
and I will believe you.' What could I do? How could I meet his challenge?
I was crushed; and he left the room triumphant.
"Does this still seem strange to you? Then put yourself
in a similar position. Suppose a person of the Fourth Dimension,
condescending to visit you, were to say, `Whenever you open your eyes,
you see a Plane (which is of Two Dimensions) and you INFER a Solid
(which is of Three); but in reality you also see (though you do
not recognize) a Fourth Dimension, which is not colour nor brightness
nor anything of the kind, but a true Dimension, although I cannot
point out to you its direction, nor can you possibly measure it.'
What would you say to such a visitor? Would not you have him locked up?
Well, that is my fate: and it is as natural for us Flatlanders
to lock up a Square for preaching the Third Dimension,
as it is for you Spacelanders to lock up a Cube for preaching the Fourth.
Alas, how strong a family likeness runs through blind and persecuting
humanity in all Dimensions! Points, Lines, Squares, Cubes, Extra-Cubes--
we are all liable to the same errors, all alike the Slaves of our
respective Dimensional prejudices, as one of our Spaceland poets has said--
`One touch of Nature makes all worlds akin.'" (footnote 1)
On this point the defence of the Square seems to me to be impregnable.
I wish I could say that his answer to the second (or moral) objection
was equally clear and cogent. It has been objected that he is a woman-hater;
and as this objection has been vehemently urged by those whom Nature's
decree has constituted the somewhat larger half of the Spaceland race,
I should like to remove it, so far as I can honestly do so. But the
Square is so unaccustomed to the use of the moral terminology
of Spaceland that I should be doing him an injustice if I were
literally to transcribe his defence against this charge.
Acting, therefore, as his interpreter and summarizer,
I gather that in the course of an imprisonment of seven years
he has himself modified his own personal views, both as regards
Women and as regards the Isosceles or Lower Classes. Personally,
he now inclines to the opinion of the Sphere (see page 86) that
the Straight Lines are in many important respects superior to the Circles.
But, writing as a Historian, he has identified himself (perhaps too closely)
with the views generally adopted by Flatland, and (as he has been informed)
even by Spaceland, Historians; in whose pages (until very recent times)
the destinies of Women and of the masses of mankind have seldom been deemed
worthy of mention and never of careful consideration.
In a still more obscure passage he now desires to disavow
the Circular or aristocratic tendencies with which some critics
have naturally credited him. While doing justice to the intellectual
power with which a few Circles have for many generations maintained
their supremacy over immense multitudes of their countrymen, he believes
that the facts of Flatland, speaking for themselves without comment
on his part, declare that Revolutions cannot always be suppressed
by slaughter, and that Nature, in sentencing the Circles to infecundity,
has condemned them to ultimate failure--"and herein," he says,
"I see a fulfilment of the great Law of all worlds, that while the wisdom
of Man thinks it is working one thing, the wisdom of Nature constrains
it to work another, and quite a different and far better thing."
For the rest, he begs his readers not to suppose that every minute detail
in the daily life of Flatland must needs correspond to some other detail
in Spaceland; and yet he hopes that, taken as a whole, his work may prove
suggestive as well as amusing, to those Spacelanders of moderate and modest
minds who--speaking of that which is of the highest importance, but lies
beyond experience--decline to say on the one hand, "This can never be,"
and on the other hand, "It must needs be precisely thus,
and we know all about it."
Footnote 1. The Author desires me to add, that the misconceptions of
some of his critics on this matter has induced him to insert (on pp.
74 and 92) in his dialogue with the Sphere, certain remarks which have
a bearing on the point in question and which he had previously omitted
as being tedious and unnecessary.
**End of The Project Gutenberg Etext of Flatland by Edwin A. Abbott**